082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

26 Chapter 1 Review of algebraic techniques
(c) h 2 +5h+1=0
(d) 0.5x 2 +3x−2=0
(e) 2k 2 −k−3=0
(f) −y 2 +3y+1=0
(g) 3r 2 =7r+2
(h) x 2 −70=0
(i) 4s 2 −2=s
(j) 2t 2 +5t+2=0
(k) 3x 2 = 50
5 Calculate the rootsofthe followingpolynomial
equations:
(a) x 3 −6x 2 +11x−6=0givenx=1isaroot
(b) t 3 −2t 2 −5t+6=0givent=3isaroot
(c) v 3 −v 2 −30v+72=0givenv=4isaroot
(d) 2y 3 +3y 2 −11y +3 = 0 giveny = 1.5 is aroot
(e) 2x 3 +3x 2 −7x−5=0givenx=− 5 2 isaroot.
6 Checkthat the given values are roots ofthe following
polynomial equations:
(a) x 2 +x−2=0
(b) 2t 3 −3t 2 −3t+2=0
(c) y 3 +y 2 +y+1=0
x=−2,1
(d) v 4 +4v 3 +6v 2 +3v=0
t=−1,0.5
y=−1
v=−1,0
Solutions
1 (a) 3 (b) −4 (c) 10 (d) 8
(e) 12 (f) 3 (g)
(i) 2 (j) 2 (k)
34
3
7
4
(h)
(l)
2 (a) 2,3 (b) −4,3 (c) 5
6
5
31
7
(d) −7,3 (e) 3,6 (f) −1,1
(g) 1,9 (h) −0.5,1 (i) −2,0.5
(j) −1, − 1 3
(m) − 1 2 , 2 3
(k) −2.5, −0.5 (l) 0.25,2
(n)
1
6 ,2
3 (a) (x+1) 2 −9=0,x=−4,2
(b) (x −3) 2 −14 = 0,x = −0.7417,6.7417
(c) (x +2) 2 −10 = 0,x = −5.1623,1.1623
(d) (x −7) 2 −59 = 0,x = −0.6811,14.6811
(e)
(
x + 5 ) 2 221 − = 0,x = −9.9330,4.9330
2 4
4 (a) −1.6180,0.6180
(b) −0.5616,3.5616
(c) −4.7913,−0.2087
(d) −6.6056,0.6056
(e) −1,1.5
(f) −0.3028,3.3028
(g) −0.2573,2.5907
(h) −8.3666,8.3666
(i) −0.5931,0.8431
(j) −2, −0.5
(k) −4.0825,4.0825
5 (a) 1,2, 3 (b) −2,1,3
(c) −6,3,4
(e) −2.5, −0.6180,1.6180
(d) −3.3028,0.3028,1.5
1.5 ALGEBRAICFRACTIONS
An algebraicfraction has the form
algebraicfraction = numerator polynomial expression
=
denominator polynomial expression
For example,
3t+1
t 2 +t+4 , x 3
x 2 +1
are all algebraic fractions.
and
y 2 +1
y 2 +2y+3