Unit-2:Polymonials (Core) - New Indian Model School, Dubai

CLASS 

X 

CBSE-i 

A L G E B R A ( C O R E ) 

POLYNOMIALS 

UNIT - 2 

Shiksha Kendra, 2, Community Centre, Preet Vihar, Delhi-110 092 India

CBSE-i 

A L G E B R A ( C O R E ) 

POLYNOMIALS 

UNIT - 2 

CLASS-X 

Shiksha Kendra, 2, Community Centre, Preet Vihar, Delhi-110 092 India

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Content 

1. Syllabus 1 

2. Scope Document 2 

3. Material Resource 5 

• 2.1 Introduction of Unit 5 

• 2.2 Value of a Polynomial at a point 7 

• 2.3 Zero of a Polynomial 7 

• 2.4 Relation between zeroes and coefficients 10 

• 2.5 Geometrical representation of zeroes of polynomial 12 

• 2.6 Reading zeroes of linear and quadratic polynomial from graph 16 

• 2.7 Division of Polynomials 18 

4. Teacher's Support Material 21 

• 2.8 Activity 1 - Warm up (W1) 22 

• 2.9 Activity 2 - Warm up (W2) 24 

• 2.10 Activity 3 - Pre Content (P1) 26 

• 2.11 Activity 4 - Content Worksheet (CW1) 26 





• 2.16 List of useful resources 29

Content 

5. Students' Support Material 31 

• Worksheet 1 - Warm up (W1 & W2) 32 

• Self assessment Rubric 1 34 

• Worksheet 2 - Pre Content (P1) 35 

• Worksheet 3 - Self assessment Rubric 2 36 

• Worksheet 4 - Content (CW1) 37 









• Worksheet 13 - Self assessment Rubric 7 

49 

• Worksheet 14 - Post Content (PCW1) 50 

• Worksheet 15 - Post Content (PCW2) 51 

6. Assessment Guidance Plan 53

SYLLABUS UNIT 2 : 

ALGEBRA (CORE)-POLYNOMIALS 

Polynomial 

Zeroes of polynomial 

Division algorithm 

Recapitulation of vocabulary of polynomials 

in one variable - coefficient, terms, degree, 

constant, linear, quadratic, cubic polynomial 

Zero of linear polynomial, zeroes of quadratic 

polynomial, finding zeroes of polynomial by 

splitting middle terms, relation between 

zeroes and coefficients. Geometrical 

representation of zeroes of polynomial, 

reading zeroes of linear and quadratic 

polynomial from graph. 

Dividend= divisor x quotient + remainder 

Division of quadratic polynomial by linear 

polynomial 

Division of cubic polynomial by linear 

polynomial 

Division of cubic polynomial by quadratic 

polynomial 

MATHEMATICS UNIT - 2 

1

SCOPE DOCUMENT 

Key terms 

1. Zeroes of a polynomial 

2. Cubic polynomial 

3. Division algorithm 

Learning Objectives 

1. To tell the possible number of zeroes for a given polynomial. 

2. To find zero of a linear polynomial algebraically. 

3. To find zeroes of a quadratic polynomial algebraically. 

4. To find zeroes of a cubic polynomial algebraically. 

5. To understand the geometrical meaning of zeroes. 

6. To read the zeroes of a polynomial from given graph. 

7. To find the polynomial when zeroes are known. 

8. To divide the linear polynomial by a linear polynomial. 

9. To divide the quadratic polynomial by a linear polynomial. 

10. To divide the cubic polynomial by a linear polynomial. 

11. To divide the cubic polynomial by a quadratic polynomial. 

12. To express the division of two polynomials using division algorithm 

2 

MATHEMATICS UNIT - 2

Extension Activities: 

Finance 

1. Find the zeroes of a cubic polynomial by the method of grouping. 

2. Find the zeroes of a bi-quadratic polynomial using division. 

Cross Curricular Links : Polynomials define simple curves in the language of mathematics so 

that they may be easily analyzed and modified. Simple curves can be combined to closely 

approximate more complicated curves. Planets, weather etc. move in curves. Mechanical 

forces, chemical and biological processes, etc. are not constant but change over space and time. 

These changes and other changes like fluctuations in the economy can be approximated by 

curves. Also, televisions, computers, phones, music players, etc. all receive signals that are 

sine waves (curves). Polynomials are the building blocks of all these sciences. 

Assessment of present value is used in loan calculations and company valuation. It involves 

polynomials that back interest accumulation out of future liquid transactions, with the aim of 

finding an equivalent liquid (present, cash, or in-hand) value. Tax and economic calculations 

can usually be written as polynomials as well. 

Electronics 

Electronics use many polynomials. The definition of resistance, V=IR, is a polynomial relating 

the resistance from a resistor to the current through it and the potential drop across it. 

Curve Fitting 

Polynomials are fit to data points in both regression and interpolation. In regression, a large 

number of data points is fit with a function, usually a line: y=mx+b. The equation may have 

more than one "x" (more than one dependent variable), which is called multiple linear. 

In interpolation, short polynomials are joined tegether so they pass through all the data points. 

For those who are curious to research this more, the name of some of the polynomials used for 

interpolation are called "Lagrange polynomials", "cubic splines" and "Bezier splines". 

Chemistry 

Polynomials come up often in chemistry. Gas equations relating diagnostic parameters can 

usually be written as polynomials, such as the ideal gas law: PV=nRT (where 'n' is mole count 

and 'R' is a proportionality constant). 


3

Formulas of molecules in concentration at equilibrium also can be written as polynomials. For 

example, if A, B and C are the concentrations in solution of OH-, H2O+, and H2O respectively, 

then the equilibrium concentration equation can be written in terms of the corresponding 

equilibrium constant K: KC=AB. 

Physics and Engineering 

Physics and engineering are fundamentally studies in proportionality. If a stress is increased, 

how much does the beam deflect? If a trajectory is fired at a certain angle, how far away will it 

land? 

Polynomials are really useful in calculating projectile paths as the physics equation for 

projectile motion is: 

s=ut + 0.5 at ² 

this is a quadratic 

Other polynomials can be derived (calculus) to find the rate of change. 

Such as if you have a distance time graph, d (t) = t² + 5 for example, you can find the velocity of 

that graph at any point in time. 

Read more : 

How are Polynomials used in Life? |eHow.com 

http://www.ehow.com/about_5479798_polynomials-used-life.html#ixzz1Inv2GL6n 

4 


2.1 Introduction of the unit: 

In previous Classes you have studied about the polynomials. Let us recall all 

concepts learnt. 

Polynomial is an algebraic expression in one variable, say x .It is the sum of two or 

n 

more terms of the form ax , where a is real number and x is a non negative integer. 

Polynomial can be written in the form 

2 n-2 n-1 n 

a 

0 

+ a1x + a2x + .....................+ an-2x + an-1x + a 

n 

x 

where a , a , a ,.......a , a , a are real number and are known as coefficients and are 

0 1 2 n-2 n-1 n 

known as non-negative integers. 

Some examples of polynomial are 

2 8 2 3 2 

x +5x+6, x +8x +15x+9, x +2x +x+4 

7 The coefficient of highest power term is known as leading coefficient. 

8 The coefficient of a term with power 0 is called a constant, 

3 

In 3x +5x+7. 7 is a constant and 3 is the leading coefficient. 

Let us revise these terms. 

What is a monomial? Polynomial with one term is known as monomial 

Examples of monomials are 

5 3 

15, 24x, 12y , -100m 

MATERIAL RESOURCES 

What is a binomial? Polynomial with two terms is known as binomial 

Examples of binomials are 

• 16y² + 25y 

• -112x³ - 23x 

• 31x + 5x² 


5

What is a trinomial? A trinomial is a polynomial with three terms 

Examples of trinomials are: 

2 

• 5x + 6x + 7 

3 2 

• -12x + 63x - 25x 

Degree of a Polynomial 

Power of variable in each term is known as degree of that term. For example for a 

3 2 

polynomial p(x) = 2x + 5x +7 x+ 6 

Term 

Degree of corresponding term 

3 

2x 3 

2 

5x 2 

7 x 1 

6 0 

The term with highest power determines the degree of a polynomial .In the above 

case the highest power is 3. So the degree of the polynomial is 3. 

If p(x) is a polynomial in x, the highest power of x in p(x) is called the degree of the 

polynomial p(x).Depending on the degree, a polynomials can be named as: 

Linear Polynomial: A polynomial whose degree is 1. 

Quadratic Polynomial: A polynomial whose degree is 2. 

Cubic Polynomial: A polynomial whose degree is 3. 

Degree Name Example 

0 constant 10 

1 linear 5x+2 

2 quadratic 5x² + 14x + 6 

3 cubic 12x³ + 3x + 7 

4 2 

4 quartic 30x + 2x + 24 

5 

5 quintic 2x + 3x + 1 

6 


2.2 Value of a polynomial at a point 

2 

Consider the polynomial p(x) = x + 5x - 3. When we put x = 1 in p(x), then we get 

2 

p(1) = 1 +5 × 1 - 3 = 3. The value '3', obtained by replacing x by 1 in p(x), is the value of 

given polynomial at x = 1. 

Now, consider p(x) = 5x+3 

p(0) = 5(0) +3 = 3 

p(1) = 5(1) +3 = 8 

p(2) = 5(2) +3 = 13 

In general, If p(x) is a polynomial in x, and if k is any real number, then the value 

obtained by replacing x by k in p(x), is called the value of p(x) at x = k, and is denoted by 

p(k). 

3 2 

What is the value of p(x) = 5x -3x -4x+1 at x= -3? 

3 2 

p(x) = 5x -3x -4x+1 

3 2 

p(-3)= 5(-3) -3(-3) -4(-3)+1 

Simplify and get the answer. 

2.3 Zero of a Polynomial 

In class 9 students have learnt the concept of finding zero of a linear polynomial. 

Definition: 

Zero of a polynomial is that value of x that makes the polynomial equal to 0. In other 

words, the number r is a zero of a polynomial P(x) if and only if P(r) = 0. 

Example 1: Consider a linear polynomial 5x + 7, 

To get its zero, we have to find the value of x, so that it becomes zero. 

Therefore 5x + 7 = 0 

⇒x = - 7/5 

Exercise: Find the zero of following linear polynomials: 

1. P(x) = 4x -3 


7

2. Q(x) = 7x + 5/3 

3. R(t) = 9t- 4 

4. P(x) = 14x + 5 

2 

Example 2: Consider a quadratic polynomial x -5x+6, 

To get its zero, we have to find the value of x, so that it becomes zero. 

2 

⇒x -5x+6 = 0 

2 

⇒x -3x-2x+6 = 0 

⇒x(x-3) -2 (x-3) = 0 

⇒(x-3)(x-2) = 0 

⇒x = 3 or x= 2 

2.3.1 How many zeroes can a polynomial have? 

Number of zeroes of a polynomial depends on its degree. A polynomial of degree n can 

have at the most n number of zeroes. 

A polynomial of degree two can have at the most two zeroes. 

2 

Polynomial x + 2x +1 has only one zero i.e. -1. 

2 

Polynomial x + 3x+2 has two zeroes -1 and -2. 

A polynomial of degree 3 can have at the most three zeroes. 

Can you write a polynomial of degree three with one zero? 

Can you write a polynomial of degree three with two zeroes? 

Can you write a polynomial of degree three with three zeroes? 

It is possible to write a polynomial when the zeroes are known. 

To write a polynomial when its zeroes are known 

In the example 2 given above you may observe that 2 and 3 are zeroes of polynomial and 

x-2 and x-3 are factors of a given polynomial. 

8 


If a given polynomial p(x) can be factorised and can be written as (x-a) (x-b) (x-c)….. 

then the polynomial will become zero if any one of its factor is zero. 

Thus we can say that if a polynomial has zeroes as a and b , it will have factors as x-a and 

2 

x-b, then the polynomial will be (x-a)(x-b)=x -(a+b)x +ab. 

Example 3: Find a polynomial of degree 3 with zeroes as -2,1,3. 

Solution: If -2 is a zero of a polynomial then x+2 is a factor of polynomial. Similarly 

other factors are x-1 and x-3.Therefore polynomial is (x+2) (x-1) (x-3). 

3 2 

Simplify to get the required polynomial p(x)= x -2x -5x +6 

2.3.2 Finding zeroes of a polynomial using constant term and leading 

coefficient 

2 

Consider the polynomial q(x)=x - 6x +5. 

Here the constant term is 5.Its factor are say p=1 and 5. 

Here the leading coefficient is 1.its only factor is say q= 1. 

Divide the factors of the constant by the factors of leading coefficient i.e. find p/q. Take 

positive and negative values of p/q. Here 1,-1 and 5,-5 are obtained. 

Insert the numbers obtained in the polynomial, 

2 

Q (1) = (1) -6(1) +5=0 

2 

Q (5) = (5) - 6(5) +5 =0 

2 

Q (-1) = (-1) -6(-1) +5≠0 

2 

Q (-5) = (-5) -6(-5) +5≠0 

You may observe that 1 and 5 are zeroes of given polynomial. 

Note : Zeroes obtained in this manner are known as Rational Zeroes. 

3 

Examples 4: Find the roots of x -3x +2=0 

Solution: Here leading coefficient= 1.Its factors are q=1 

Constant term= 2. Its factors are p = 1 and 2 

So, p/q= 1 and 2 


9

Take both positive and negative values of p/q=1,-1,2,-2 

3 

P(1)= (1) - 3(1)2 +2=0 

3 

P(-1)= (-1) - 3(-1)2 +2≠0 

3 

P(2)= (2) - 3(2)2 +2≠0 

3 2 

P(-2)= (-2) - 3(-2) +2≠0 

Therefore x=1 is a zero of the given polynomial. But the given polynomial is of degree 3, 

so it can have at the most three zeroes. From above method we are able to find only one 

zero. Other two zeroes in this case can be determined by dividing the given polynomial 

by x-1 as x-1 is one of the factors of given polynomial. 

We will first learn the division algorithm and then will learn further to find all zeroes of 

cubic polynomial. 

Example 5. Let P(x) = 5x³ - 4x² + 7x - 8. Find P (1).What do you conclude about the 

factors of P(x)? 

2.4 Relationship between zeroes and coefficients of a quadratic polynomial 

What is a quadratic polynomial? 

2 

A polynomial ax +bx+c, a ≠0, a, b, c, are real numbers is called a quadratic polynomial. 

Consider a quadratic polynomial 5x²+ 6x +1. To get its zero, we write 5x²+ 6x +1 = 0. This 

means we have to find the value (s) of the variable x which makes the given expression 

0. 

By splitting the middle term, we get 5x²+ 5 x +x +1 = 0. 

This means, 5x(x+1) + 1(x+1) = 0 

⇒(5x+1)(x+1) = 0 

⇒x = -1/5 or x = -1 

Therefore, the two zeroes are -1/5 and -1 

Now, find the sum of zeroes and product of zeroes 

10 


Sum of zeroes = 

- 1 5 -1= - 6 5 

Observe the coefficient of x and the coefficient of x² terms. 

We find that Sum of zeroes = - 

Coefficient of x 

coefficient of x² 

Now, find the product of zeroes 

Product of zeroes = 

- 1 5 x -1= 1 5 

2 

Observe the constant term and the coefficient of x terms. 

We find that Product of zeroes = 

Coefficient term 


In a quadratic polynomial ax²+bx+c, a 0, a, b, c, are real numbers 

Sum of zeroes = - 

Product of zeroes = 





Example: Consider a quadratic polynomial 2x²-8x + 6 

By splitting the middle term we get, 

2x²-8x+6 = (2x-2)(x-3) 

= 2(x-1)(x-3) 

Zeroes of this polynomial are given by x-1=0 and x-3=0 

x=1 and x=3 

Sum of zeroes = 1+3 = 4 = -(-8)/2 = - 



Product of zeroes =( 1) (3) = 3 = 6/2 = 




11

Now, let us learn to write a quadratic polynomial when sum and product of its zeroes is 

given. 

Remember: A quadratic polynomial whose sum of zeroes is S and product of zeroes is P 

is given by x² -Sx +P 

Example: Write a quadratic polynomial whose sum of zeroes is 5 and product of zeroes 

is 10 

Here Sum of zeroes = 5 and Product of zeroes= 10 

Using the formula, required polynomial is x² - 5x + 10 

Note: If αand βare zeroes of a quadratic polynomial then the required polynomial is 

given by (x- α) (x- β) 

2.5 Geometric Representation of Zero of a Polynomial 

What are the x-intercept and y-intercept of a graph? 

(0,b) 

y-intercept 

(-a, 0) 

x-intercept 

The x-intercept is that value of x where the graph crosses or touches the x-axis. At the x- 

intercept on the x-axis y = 0. 

The y-intercept is that value of y where the graph crosses the y-axis. At the y-intercept, 

x = 0. 

What is the relationship between the zero of a polynomial and the x-intercepts of its 

graph? 

The zeroes are the x-intercepts! 

12 


The zeroes of x² - x - 6 are -2 and 3. Therefore, the graph of y = x² - x - 6 will cross the 

x - axis at -2 and 3. 

The graph of a linear polynomial of the form ax + b, a not equal to zero is a straight line 

which intersects the x-axis at exactly one point. The value of x coordinates of the point 

where the line meets x- axis is zero of the polynomial. 

Graph of a linear Polynomials 

Consider the following graph 


13

This is a graph of linear polynomial 2x+3. It intersects the x-axis at point (-1.5, 0). 

So, the linear polynomial 2x+3 has one zero, given by -1.5 (x-coordinate of point where 

line cuts the x-axis) 

Exploring graphs of Quadratic polynomial 

2 

• A quadratic polynomial is given by ax +bx+c, a ≠0, a,b,c, are real numbers. 

• The degree of a quadratic polynomial is 2. 

• Graph of a quadratic polynomial is a parabola. 

• There are at most two zeroes of a quadratic polynomial. 

Observe the following graphs. 

This is a graph of a quadratic polynomial which do not intersect the x- axis at any of the 

points, so there is no zero of this polynomial. 

14 


In the second graph, the curve touches the x-axis at one point, so it has 1 zero. What is 

that zero? 

In the following graph, the curve cuts the x-axis at two points, so the given quadratic 

polynomial has two zeroes. Write the two zeroes from the graph. 

Note: The zero of a quadratic polynomial is the value of x- coordinate of a 

point where the graph of the polynomial cuts the x-axis. It has at most 2 zeroes 


15

2.6 Observing the number of zeroes of a polynomial from its graph 

(i) 

The graphs of y = p(x) are given in following figure, for some polynomials p(x). Find the 

number of zeroes of p(x), in each case. 

(ii) 

Observation: The graph of y=p(x) does not intersect the x-axis at any point. This means 

it has no zero. 

(iii) 

Observation: The graph of y=p(x) intersects the x-axis at one point. This means it has 

one zero. 

Observation: The graph of y=p(x) intersect the x-axis at three points. This means it has 3 

zeros. 

16 


(iv) 

Observation: The graph of y=p(x) intersect the x-axis at two points. This means it has 2 

zeros. 

(v) 

Observation: The graph of y=p(x) intersect the x-axis at four points. This means it has 4 

zeros. 

(vi) 

Observation: The graph of y=p(x) intersect the x-axis at three points. This means it has 3 

zeros. 


17

2.7 Division of polynomials 

In previous classes students have learnt the division of integers. Also, they have the 

knowledge of Euclid's division lemma i.e. dividend = divisor x quotient + remainder. 

Let us see the process of division of polynomials. 

First let us divide 5x³ - x³ + 6 by x - 4. 

We write it as shown below 

x - 4 5x³ - x² + 0x + 6 

We need to have the terms written down with the exponents in decreasing order and to 

make sure we don't make any mistakes we add in any missing terms with a zero 

coefficient. 

Now we we need to multiply x - 4 to get the first term in first polynomial. In this case 

that is 5x². So multiply x - 4 by 5x² and subtract the results from the first polynomial. 

5x² 

x - 4 5x³ - x² + 0x + 6 

- (5x³ - 20x²) 

19x² + 0x + 6 

The new polynomial is called the remainder. We continue the process until the degree 

of the remainder is less than the degree of the divisor, which is x - 4 in this case. So, we 

need to continue until the degree of the remainder is less than 1. 

Recall that the degree of a polynomial is the highest exponent in the polynomial. Also, 

recall that a constant is thought of as a polynomial of degree zero. Therefore, we'll need 

to continue until we get a constant in this case. 

Here is the rest of the work for this example. 

5x² + 19x + 76 

x - 4 5x³ - x² + 0x + 6 

- (5x³ - 20x²) 

19x² + 0x + 6 

- (19x² - 76x) 

76 + 6 

- (76x - 304) 

310 

18 


What is division algorithm for polynomials? 

Division algorithm states that : Given a polynomial P(x) with degree at least 1 and any 

number R there is another polynomial Q(x), called the quotient, with degree one less 

than the degree of P(x) and a number R, called the remainder, such that, 

P (x) = (x - r) Q (x) + R 

Division algorithm: Dividend = Divisor x Quotient + Remainder 

Example Divide x³ -3x² +5x -3 by x² -2 

(I) p(x) = x³ - 3x² + 5x - 3 

q(x) = x² - 2 

x - 3 

x² - 2 x³ - 3x² + 5x - 3 

x³ - 2x 

- + 

- 3x² + 7x - 3 

- 3x² +6 

+ - 

7x - 9 

Quotient = x - 3 

Remainder = 7x - 9 

2 4 3 2 

Example To check whether x +3x+1 is a factor of 3x + 5x -7x +2x+2. 

3x² - 4x + 2 

4 

x² + 3x + 1 3x + 5x³ - 7x² + 2x + 2 

4 

3x + 9x³ + 3x² 

- - - 

- 4x³ - 10x² + 2x + 2 

- 4x³ - 12x² - 4x 

+ + + 

2x² + 6x + 2 

2x² + 6x + 2 

0 


19

Since the remainder is 0, 

4 

Hence, x² + 3x + 1 is a factor of 3x + 5x³ - 7x² + 2x + 2 

Discussion: How many zeroes a cubic polynomial has? Yes, you are right. A cubic 

polynomial has at most 3 zeroes. 

You have learnt how to find rational zero of a cubic polynomial. For the problem 

discussed earlier we were able to found one zero and mentioned that rest of the zeroes 

we will be able to find after learning Division Algorithm. Now, we will use the division 

of polynomials process to do the same. 

Example: If one zero of a cubic polynomial x³ - 3x² - x+3 is 1, then find the other zeroes. 

In this problem, it is given that one zero is 1, this means (x-1) is a factor of given 

polynomial. 

We will divide the given polynomial by (x-1). 

On dividing we get x³ - 3x² - x+3 = (x-1)(x²-2x-3) +0 

But x² - 2x - 3 can be factorised as (x+1) (x-3) 

.: x³ - 3x² - x + 3 = (x-1) (x²-2x-3) + 0 

= (x-1) (x+1) (x-3) + 0 

⇒Factors of the polynomial x³ - 3x² - x + 3 

are (x-1) (x+1) (x-3) 

Note : Find all zeroes of polynomial x³ - 3x + 2 = 0 

You already know that one zero of this polynomial is 1. 

20 


Teachers' 

Support 

Material 


21

2.8 Activity 1- Warm up (W1) 

Description: This is a starter activity to engage students in learning. Show the image 

given below and ask the students to read the words one by one. Tell them we will play a 

game using these terms. 

Pre Preparation: 

Prepare paper slips on which various terms and 

statements are written. Put them in a bowl and 

ask each student to pick up one slip. 

22 


monomial 

A binomial having degree 4 

A trinomial having degree 4 

A constant polynomial 

A cubic polynomial 

A quadratic polynomial 

A trinomial whose constant 

term is 1 

Execution: Each student will have to say one sentence about the written terms or give 

examples in support of statement on the slip. 


23

2.9 Activity 2- Warm up (W2) 

Specific Objectives: 

• To explore an algebraic expression and find whether it is a polynomial or not? 

• To find If the given expression is a polynomial then it is a polynomial in one variable 

or not? 

• To find If the given expression is a polynomial then observe its degree and count the 

number of terms 

• To name the given polynomial according to its degree. 

• To name the given polynomial according to the number of terms in it. 

Description: This is a brainstorming warm up activity. Write any algebraic 

expression on the board. Students will be asked to comment on the expression. They 

will brainstorm on various aspects 

• Is it a polynomial? Why and Why not? 

• If yes, then what is the degree? What will you call it on the basis of degree? 

• How many terms are there? What will you call it on the basis of number of terms? 

• What are the coefficients of various terms? 

Pre preparation: 

Make a list of various algebraic polynomials. 

Some of them are not to be polynomials. 

Prepare a list of points for brainstorming. 

Terms: Monomial, Binomial, Trinomial, Polynomial, Degree, Quadratic polynomial, 

Cubic Polynomial, Coefficients 

24 


Execution : 

Consider the given algebraic expression 

4x³ + 3x² - 5x - 6 

Ask the students to write observations on the following statements 

1. The given algebraic expression is a polynomial 

2. It is a polynomial in one variable 

3. The coefficient of x is 5 

4. Its constant term is 6 

5. Its degree is 5 

6. It is a cubic polynomial 

7. It has 4 terms 

8. The coefficient of x² is 3 

9. If we remove -5x term then it will become a trinomial 

10. If we remove any of one term, then it will become a trinomial 

11. If we add one more term having degree 4 then it will become a polynomial of degree 4. 


25

2.10 Activity 3 Pre Content (P1) 


• To recall the concept of value of a polynomial at a given point 

• To practice to find the value of a polynomial at a given point 

• To revise the concept of zero of a linear polynomial 

Description: Students have studied in class 9 to find the value of a polynomial at a given 

point. Also, they have learnt to calculate zeroes of a linear polynomial. For example, to 

calculate the zero of linear polynomial p(x) = 3x -2, put 3x-2 = 0. So, the zero of p(x) = 3x - 

2 will be 2/3. 

Prepare a worksheet for students for testing the previous knowledge based on two 

concepts. 

2 Finding the value of a polynomial at a given point 

3 Zeroes of a linear polynomial 

Execution: Ask the students to solve the worksheet. Discuss orally the important 

points. Let students speak the method of finding zero of a linear polynomial. 


Description: 

2.11 Activity 4 Content Worksheet (CW1) 

• to be able to write linear and quadratic polynomials 

• to be able to find the zeroes of a linear polynomial 

• to be able to find the zeroes of a quadratic polynomial by splitting the middle term 

After learning to find the zeroes of a polynomial, students will be provided with 

content worksheet CW1. It is a 20 minutes task in which students will use their 

knowledge and solve the exercise given in the worksheet. 

26 


Follow up: 

Revise important points learnt till now. Ask the students to make a mind map. 


2.12 Activity5 Content Worksheet (CW2) 

• To find the zeroes of a quadratic polynomial and verify the relationship between zeroes 

and coefficients 

• To write a quadratic polynomial when its zeroes are given 

• To write a quadratic polynomial when sum and product of its zeroes are given 

Description: After learning how to find the zeroes of a quadratic polynomial, students 

will be given a worksheet on finding a relationship between zeroes and coefficients of a 

quadratic polynomial. Students will use the knowledge of splitting the middle term 

concept and find the zeroes of a given quadratic polynomial. They will further verify 

the relationship between zeroes and coefficients of the polynomial. 



• To practice the skill of drawing graphs of a linear polynomial on a graph paper 

• To observe the graph of a polynomial and tell the number of zeroes 

• To tell the value of zero(s) of a polynomial from its graph 

Description: Teacher will ask the students to write a linear polynomial and draw its 

graph. Each student will make the table of values and plot the points on a graph paper. 

Students will be then asked to observe the points where the graph intersects the x- axis 

(if any). 

Let students verify that the abscissa of the point where the graph intersects the x- axis is 

the required zero of the given polynomial. 


27



To observe the number of zeroes of a polynomial from its graph. 

Description: 

Students will be asked to observe the graphs of polynomials. They will be asked. 

(i) Is the graph intersecting x-axis at any point? 

(ii) At, how many points, the graph in intersecting x-axis? 

Hence, how much zeroes, the polynomials have? 


Description 


To divide a polynomials of degree more than 1 by a linear polynomial. 

To find the quotient and the remainder. 

To understand the division algorithm to establish that 

Dividend = Divisor x Quotient + Remainder. 

Students will be given two polynomials p(x) and g(x) and will be asked. 

1. Which of these polynomials can be divided by the other? 

2. What will be the degree of quotient? 

3. What will be the degree of the remainder? 

Take different examples of p(x) and g(x) and find q(x), the quotient and r(x), the 

remainder. 

28 


2.16 Useful Resources 

(i) 

http://www.sosmath.com/algebra/factor/fac01/fac01.html 

(On this web page the process of long division of polynomials is explained step by step) 

(ii) 

http://www.mathsisfun.com/algebra/polynomials-division-long.html 

(On this web page, some videos are inserted for the explanation of long division in 

polynomials) 

Useful Videos 

i. Note: If the students find problem in splitting the middle term of a quadratic 

polynomial then they may watch Video: Splitting the middle term 

http://www.youtube.com/watch?v=07IenNnS3Xs 

ii. http://www.youtube.com/watch?v=FsotIB0Usvw 

iii. 

Dividing a quadratic polynomial by a linear polynomial 

http://www.youtube.com/watch?v=l6_ghhd7kwQ&feature=relmfu 


29

Students' 

Support 

Material 


31

Student's Worksheet-1 

Warm up (W1) 

Name of student______________________________________ 

Date__________________ 

Consider the given algebraic expression 

3 2 

4x + 3x - 5x - 6 

Give your comments on the following statements: 

• The coefficient of x is 5 

_____________________________________________________________ 

• This is a polynomial 

_____________________________________________________________ 

• Its constant term is -6 

_____________________________________________________________ 

• Its degree is 5 

_____________________________________________________________ 

• It is a cubic polynomial. 

_____________________________________________________________ 

• It has 4 terms 

_____________________________________________________________ 

• The coefficient of x² is 3 

_____________________________________________________________ 

32 


• If we remove -5x term then it will become a trinomial 

_____________________________________________________________ 

• If we remove any of one term, then it will become a trinomial 

_____________________________________________________________ 

• If we add one more term having degree 4 then it will become a polynomial of degree 4. 

_____________________________________________________________ 

Student's Worksheet - 1A 

Warm up (W2) 

Name of student________________________________________ 

Date__________________ 

Understanding of terms along with their meaning 


33

Self Assessment Rubric - 1 

Term 

Polynomial 

Monomial 

Binomial 

Trinomial 

Linear Polynomial 

Quadratic Polynomial 

Cubic Polynomial 

Quartic Polynomial 

Coefficient of terms 

Degree of polynomial 

Knows the meaning Aware of the term but 

do not know the 

meaning 

Do not know 

what it is? 

34 



Pre content (P1) 

Name of student_________________________________________ 

Date___________________ 

Choose the correct answer. 

1. The zero of the polynomial P(x) = x+5 is 

(i) 5 (ii) -5 (iii) 0 (iv) 1 

3 

2. The remainder when x +1 is divided by x+1 is 

(i) 1 (ii) 2 (iii) -1 (iv) 0 

3 2 

3. If x-1 is a factor of 4x +3x -4x+K, then value of K is 

(i) 3 (ii) -3 (iii) 0 (iv) -2 

2 

4. The value of the polynomial 5x-4x +3 at x= -1 is 

(i) 6 (ii) -1 (iii) -6 (iv) 3 

Fill the table given below 

Polynomial 

3 

P(x)= 4x -5x+2 

2 

P(t) = 5t +3t -8 

Value 

P(2)= 

P(1)= 

P(y)=7y² + 2u - 1 P(½) = 

4 3 

P(r) = 4r + 7y + 2 P(0) = 

P(y) = 1 - ½y³ P(-1) = 

P(r)= ar³ + 1 P(1) = 


35

Self Assessment Rubric 2 


Pre Content (P1) 

Name of student____________________________ 

Date__________________ 

Rate your knowledge according to the given scale. 

Skill 

Able to find the zero of a 

linear polynomial 

Able to apply Remainder 

theorem 

Able to apply factor 

theorem 

Able to find value of a 

polynomial at a given point 

36 



Content Worksheet (CW1) 

Name of student_____________________________________________ Date______________ 

1. Write two examples each of a linear polynomial and a quadratic polynomial. 

Type of polynomial 

Examples 

Linear 

Quadratic 

2. Find the zeroes of following linear polynomials. 

a) P(x) = 7x +3 

b) Q(t) = 3t -2 

3. Take any three linear polynomials. Find the zeroes. How many zeroes a linear 

polynomial has? 

___________________________________________________________________________ 

___________________________________________________________________________ 

___________________________________________________________________________ 

___________________________________________________________________________ 

___________________________________________________________________________ 

___________________________________________________________________________ 


37

4. Factorise the following quadratic polynomials using the splitting the middle term. Find 

the zeroes. 

2 

a) 5x -6x + 1 

2 

b) t -2x-8 

2 

c) 4m -4m+1 

38 






Date__________________ 


Skill 

Able to write a linear/ 

quadratic/cubic polynomial 

Able to find zeroes of a 



quadratic polynomial 


39



Name of student_____________________________________________ Date______________ 

Assignment: 

Q1 

Find the zeroes of the following quadratic polynomials and verify the relationship 

between zeroes and coefficients. 

(a) x² + 8x + 12 

(b) x² + 3x - 4 

(c) x² - 7x + 10 

(d) y² - 4 

Q2 

Q3 

Q4 

Find a quadratic polynomial each with the given numbers as the sum and product of its 

zeroes respectively: 

3 3 

(a) 3 and 4 (b) -2 and (c) - and 0 (d) -2 and 3 

2 2 

Find a quadratic polynomial with -1 as zero. 

Find a cubic polynomial with zeroes as 2, -2½. 

Q5 Find a cubic polynomial with zeroes 1, 1, 2. 

Q6 Find a cubic polynomial with zeroes -1, -1, -1. 

40 






Date__________________ 


Skill 

Able to verify the relation 

b e t w e e n z e r o e s a n d 

coefficient 

Able to write a quadratic 

polynomial when its zeroes 

are given 

Able to write a quadratic 

polynomial when sum and 

product of its zeroes is 

given 


41




Date__________________ 

1. Draw the graph of y = 2x + 5. 

X 

Y 

Find the points where the straight line cuts the x- axis and the y- axis. 

The straight line cuts the x- axis at …………………….. 

The straight line cuts the y- axis at ……………………. 

Now, to get zero, 2x + 5 = 0 

⇒x = -5/2 

Observe that the graph of y = 2x + 5 intersects the x- axis at the point ( -5/2 , 0). 

Note: The zero of the polynomial 2x + 5 is the x-coordinate of the point i.e. - 5/2 where the 

graph of y = 2x + 5 intersects the x-axis. 

2. Draw the graph of following linear polynomials. Find the zero from the graph. How 

many zeroes a linear polynomial has? Verify you answer. 

a) 5x-3 

b) 2x+6 

c) 6x-2 

d) 2x-4 

42 


3. The graph of y = p(x) are drawn below. Mark the number of zeros in each case: 

a) b) 

c) d) 

e) f) 


43





Date__________________ 


Skill 

Able to draw the graph of a 



linear polynomial from its 

graph 


polynomial from its graph 

44 





Date__________________ 

• A polynomial of degree n has at most n zeroes. 

• Geometrically the zeroes of a polynomial are x coordinates of points where the graph of 

polynomial cuts/touches the x-axis. 

The zeroes of a polynomial p(x) are precisely the x-coordinates of the points, where the 

graph of y = p(x) intersects the x -axis. 

Observe the given graphs and fill the respective rows. 

S. No. Graph of y = f(x) 

1. Graph of a 

linear 

polynomial 

Number of Zeroes 

Zeroes 

2. Graph of a 

quadratic 

polynomial 


45

S. No. Graph of y = f(x) 

3. Graph of a 

quadratic 

polynomials 

Number of Zeroes 

Zeroes 

4. Graph of a 

quadratic 

polynomial 

46 






Date__________________ 


Skill 

Able to find number of 

zeroes of a polynomial 

from its graph 


polynomial from its graph 


47




Date__________________ 

Exercise: 

1. Apply the division algorithm to find the quotient and the remainder on dividing p(x) by 

q(x) 

3 2 

I. p(x)= x +15x +48x, q(x)= x+8 

3 2 

ii. p(x) = 14x -5x +9x-1, q(x) = 2x-1 

2. Check whether the first polynomial is a factor of the second polynomial by dividing the 

second polynomial by the first polynomial: 

2 

a) (x-1), 5x -3x+2 

2 

b) (x+1), 2x +5x+3 

3 2 

3. On dividing x -3x +5x -3 by a polynomial g(x), the quotient is (x-3) and the remainder is 

(7x-9). Find g(x). Write the formula used. 

3 2 

4. Find the value of a and b so that 1, -2 are the zeroes of the polynomial x + 10x + ax + b. 

5. Divide the polynomial p(x) by q(x) in each of the following cases and find the quotient 

and remainder: 

3 

a) p(x) = 7x - 5x + 8 q(x) = x - ½ 

b) p(x) = 6x² - 5x² + 4x + 3 q(x) = 4x² + 1 

c) p(x) = 16x³ - 4x² + 2x + 7 q(x) = 2x - 1 

48 






Date__________________ 


Skill 

Able to divide a quadratic polynomial by 

a linear polynomial 

Able to divide a cubic polynomial by a 


Able to divide a cubic polynomial by a 


Able to verify division algorithm for 

polynomials 

Able to find zeroes of a polynomial using 

division algorithm 


49


Post Content Worksheet (PCW1) 

1. Fill in the blanks 

i. Polynomials of degrees 1, 2 and 3 are called………….. , …………….. And cubic 

polynomials respectively. 

2 

ii. A…………… polynomial in x with real coefficients is of the form ax + bx + c, where 

a, b,c are real numbers with a?0. 

iii. The zeroes of a polynomial p(x) are precisely the …………………… of the points, 

where the graph of y = p(x) intersects the x -axis. 

iv. A quadratic polynomial can have at most ………….. zeroes and a cubic polynomial 

can have at most …………… zeroes. 

v. If and βare the zeroes of the quadratic polynomial ax² + bx + c, then 

a. α+ β= …………… and αβ= ………………… 

vi. The division algorithm states that given any polynomial p(x) and any non-zero 

polynomial g(x), there are polynomials q(x) and r(x) such that 

p(x) = g(x) q(x) + r(x),where r(x) = 0 or degree r(x) ……………. degree g(x). 

2. Find the quadratic polynomial, the sum and product of whose zeroes are 

1) ½, -2 

2) -3, -7 

3) 5+√3 & 5-√2 

3. Find the zeroes of the quadratic polynomial p(x) =t ² - 15 

4. How many maximum zeroes can a polynomial of degree two has? 

5. What is the zero of the polynomial ax + b=0, a≠0? 

6. Find the sum & the product of the zeroes of the polynomial 6x²-x-2. 

50 


7. Give examples of polynomials f(x), g(x), q(x) & r(x) which satisfy the division algorithm 

1) deg r(x)=0 

2) deg f(x)=deg q(x)=2 

8. Find the quadratic polynomial whose one zero is 5 & product of zeroes is 30. 

9. The linear polynomial ax + b, a ? 0, has exactly one zero, namely, the__________ of the 

point where the graph of y = ax + b intersects the x-axis. 

10. Divide the polynomial p(x) by the polynomial g(x) and find the quotient and remainder 

in each of the following: 

3 2 2 

(i ) p(x)=x -3x +5x-3,g(x)=x -2 

3 2 2 

(ii) p(x) = x - 3x + 4x + 5, g(x) = x + 1 - x 


Post Content Worksheet (PCW2) 

1. Express the polynomial given in column I in the form of and match with the 

coefficients of their terms in Column II 

COLUMN I 

COLUMN II 

x² + 6x a = 1 b = 0 c = 6 

x² + 6 a = 0 b = 1 c = 6 

x+ 6 a = 1 b = 6 c = 0 

2. Match the polynomials in column I with their zeros in column II 

COLUMN I 

COLUMN II 

x² - 5x - 6 α= -6 β= 1 

x² + 5x - 6 α= -6 β= 1 

x² - 7x + 6 α= -6 β= 1 

x² + 7x + 6 α= -6 β= 1 


51

3. Fill in the appropriate boxes: 

Dividend Divisor Quotient Remainder 

x² + 3x + 4 x + 1 2 

x² + 7x + 7 x + 5 1 

x² - 5x + 6 x - 1 -2 

x² - 3x - 12 x - 5 x + 2 

4. Fill in the blanks:- 

i) The zeroes of the polynomial ________________ are -3 and -4. 

ii) The remainder is ________________ when x³ + 5x² + 3x - 2 is divided by x + 1 . 

iii) 

If the zeroes of the polynomial are 5 and 6 then the co-efficient of is _________. 

iv) 

If the zeroes of the polynomial are opposite in sign then the co-efficient of x is 

______. 

v) The sum and product of zeroes of the polynomial ______________ are -2 and 

vi) 

The polynomial x² + 10x + 25 has______________ roots. 

-3 

2 

vii) 

If a polynomial of degree five is divided by a quadratic polynomial then the 

degree of the quotient polynomial is _______________. 

viii) The quotient polynomial is ______________ if the two zeroes of the polynomial 

x³ + 5x² - 4x + 5 are 2 and -2. 

52 


Assessment guidance plan for teachers 

With each task in student support material a self -assessment rubric is attached for students. 

Discuss with the students how each rubric can help them to keep in tune their own progress. 

These rubrics are meant to develop the learner as the self motivated learner. 

To assess the students' progress by teacher two types of rubrics are suggested below, one is for 

formative assessment and one is for summative assessment. 

Suggestive Rubric for Formative Assessment (exemplary) 

Parameter 

Mastered 

Developing 

Needs motivation 

Needs personal 

attention 

Factorisatio 

n of 

polynomial 

Able to factorise 

quadratic 

polynomial by 

splitting of middle 

term 


quadratic 

polynomial using 

formula 

Able to find atleast 

one factor using 

leading coefficient 

and constant term 

Able to use division 

of polynomials to 

find other factor 

Able to find out the 

factors of cubic 

polynomial by 

grouping them 

Able to find out 

atleast one factor by 

use of leading 

coefficients and 

constant term 

Able to find other 

factors using 

division of 

polynomials and 

repeating the same 

procedure to get all 

factors 

From above rubric it is very clear that 


quadratic 

polynomial by 


term 


quadratic 


formula 





Able to use 

division of 

polynomials to find 

other factor 

Able to find out the 

factors of cubic 

polynomial by 

grouping them 


atleast one factor 

by use of leading 


constant term 

Able to find other 

factors using 

division of 

polynomials but 

not able to repeat 

the same procedure 

to get all factors 


quadratic 

polynomial by 


term 

Not able to 

factorise quadratic 


formula 





Not able to use 

division of 

polynomials to 

find other factor 

Not able to find 

out the factors of 

cubic polynomial 

by grouping them 



by use of leading 


constant term 


other factors using 

division of 



procedure to get 

all factors 

Not able to factorise 

quadratic 

polynomial by 


term 

Not able to factorise 

quadratic 


formula 



using leading 

coefficient and 

constant term 

Not able to use 

division of 

polynomials to find 

other factor 

Not able to find out 

the factors of cubic 

polynomial by 

grouping them 

Not able to find out 

atleast one factor by 

use of leading 


constant term 


other factors using 

division of 



procedure to get all 

factors 


53

• 

• 

• 

• 

Learner requiring personal attention is poor in concepts and requires the 

training of basic concepts before moving further. 

Learner requiring motivation is able to do a lot but stuck up with division of 

polynomials. He can be trained by peer trainers or by doing remedial 

worksheets. 

Learner who is developing is able to almost all type of problems but needs more 

practice to solve cubic polynomials and to understand how the procedure can be 

repeated. 

Learner who has mastered the skill of factorisation of polynomials can be given 

higher order polynomials for factorisation. 

Teachers' Rubric for Summative Assessment of the Unit 

Parameter 

Factorisation of 

polynomial 

algebraically 

Finding zeroes 

of polynomial 

Division 

algorithm 

5 

• Able to factorise 


• Able to factorise cubic 

polynomial 

• Able to factorise 

biquadratic polynomial 

• Able to find zeroes of 

polynomial by 

factorising it. 

• Able to read zeroes 

from graph of 

polynomial 

• Able to find zeroes of 

polynomial using the 

relation between 

coefficients of 

polynomial and its 

zeroes. 

• Able to divide linear 

polynomial by a linear 

polynomial. 

• Able to divide 


by a linear polynomial. 

• Able to divide cubic 


polynomial. 

• Able to divide cubic 

polynomial by a 

quadratic polynomial. 

1 

• Not able to factorise 



cubic polynomial 


biquadratic polynomial 

• Not able to find zeroes of 

polynomial by 

factorising it. 

• Not able to read zeroes 

from graph of 

polynomial 

• Not able to find zeroes of 

polynomial using the 

relation between 

coefficients of 

polynomial and its 

zeroes. 

• Not able to divide linear 


polynomial. 

• Not able to divide 

quadratic polynomial by 

a linear polynomial. 

• Not able to divide cubic 


polynomial. 

• Not able to divide 

cubic polynomial by a 

quadratic polynomial. 

54 


CENTRAL BOARD OF SECONDARY EDUCATION 

Shiksha Kendra, 2, Community Centre, Preet Vihar, 

Delhi-110 092 India

Unit-2:Polymonials (Core) - New Indian Model School, Dubai

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