f;a~tors and &r~tast ~omraon Factors

). 481)adoring by grouping (P. 482)greatest common factor (GCF) (p. 476)pedeCt square trinomials (P. 508)prime factorization (P. 475)prime number (p. 474)State whether each sentence is true or false. If false, replace the underlinednumber to make a true sentence. ~ge~.~-’~ ,’~--1. The number 27 is an example of a rp_rp_rp_rp_rp_rp_r~_~e number.2.~x is the greatest common factor (GCF) of i2x 2 and 14xy.~ 3. 6__6_6 is an example of a perfect square.prime polynomial (P. 497)Square Root Property (p. 51 I)Zero Product Properbl (P. 483)word orSee pages474-479.f;a~tors and &r~tast ~omraon FactorsConcept Summa~/¯ The greatest commOn factor (GCF) of two or more monomials is theproduct of their commOn prime factors.Find the GCF of 18x y and 48xy ¯15x2y ~Factor each number.45xy 2. y Circle the common prime factors.The GCF is 3 ’ 5 ¯ x ’ y or 15xy.Exercises Find the prime factorization of each integer.See Examples 2 and 3 on page 475. 13. 150 ,a~. ~" ~ "~ - ~Find the GCF of each set of monomials. See Example 5 on page 476.: 17. 35,30 ~ ~ 21. 20n 2,25np~ 22. 60xy,35xz ~~20. 16tort, 30m 2r ........................................... Chapter 9 s*udy Guide and Reviewalgebra 1 com/vocabulaw_rewe~515

See pages481-486,Fa~torin9 Using tha Pistributiva l~ropartyConcept Summary¯ Find the greatest common factor and then use the Distributive Property.¯ With four or more termS, try factor~g by grouping.Factoring by Grouping: ax + bx + ay + lry = x(a + b) + y(a + b) = (a +~Factoring can be used to solve some equations.Zero Product Property: For any rea! numbers a and b, if ab = 0, theneither a = 0, b = 0, or both a and b equal zero.Factor 2x 2 - 3xz - 2xy + 3!~z.2X 2 -- 3XZ -- 2xy + 3~d z = ( 2x2 Group terms with common factors.- 3xz) + (-2xy + 3yz)Factor out the GCF from each grouping.= x(2x - 3z) - y(2x - 3z)Factor out the common factor ~x - 3z.=- (X -- y)(2x -- 3z)olynomiaL See Examples 1 and 2 on pages 481 and 482.¯, 27. 4rs + 12ps + 2mr j- 6mP~l~ ~(; ~~-k .3. e,~’~3 ~ our solutions. See~amples2and~°npages~and~"olve eac~ equahon. Check y30. (3n + 8)(2n - o) =. v ¯~a~torin9 Trinom~als:See ~ages Concept Su~ma~489-494. ¯ Factorb~g x ~ + bx + c: F~d m and n whose sum is b an~ whose product is~en write x 2 + bx + c as (x + m)(x + n).Solve a 2 - 3a - 4 = 0. Then check the solutions.a~-3a-4=0(a+~)(a-4)=0a+l=0 or a -4=0a = -1 a=4Original equationFactor.Zero Product PropertySolve each equation.The solUtiOn set is {-1, 4}.See Examples ! 4 on pages 490 and 491. l’ ~ ¯ ~ trinomi al - ¯ "~ ’’-~Exercises Factor.each ~,’"2 ,~ "a.4(-~oV’~.’~ 34" ~b2+Sb-6 ~ ~)~2 ,2 + 7u + 12 Q-~ ~’~J 33. x - 7~ - ~ ~2 " ~ ~ m 2 -- 4ran = 32n~9 r 2 ~u’" ~ ~ ~35. _. )

Fa~torin9 Trinornidls: ax + bx +See pa~eS : Concept Summa~~s-~oo. ¯ Factor~g ax 2 + bx + c: Find m ~d n whose product is ac and whose sum is b. Then,write as ax 2 +mx + nx + c and use factoring by grouping.Factor 12x 2 + 22x - 14...... mE ’~ 12x 2 + 22X -- 14 = 2(6x 2 + 11X - 7). In the new trinomia!,aFi actorr=st~b = out me11 and c = -7. Since b is positive, m + n ~s poslhve. Smce c ~s negative,mn is negative. So either m or n is negative, but not both. Therefore, make a list ofthe factors of 6(-7) or -42, where one factor in each pair is negative. Look for apair of factors whose sum is I1.Factors of -42 Sum of Factor__s-1, 42 411, -42 -41-2, 21 192, -21 -19--3, 14 11 The correct factors are -3 and 14.Solve each equation. Check your solutions.48. 3a 2-13a+14=0;age 497.+ 2x = 24 IFa~torin9vquar~sSee pa~es :. Concept Summan/501-50~. :~ ¯ Difference of Squares: a 2 - b2 = (a + b)(a - b) or (a - b)(a + b). Sometimes it may be necessary to use more than one factoring techniqueor to apply a factoring tect’mique more than once.Factor 3x 3 -- 75x.3x 3 -- 75x = 3x(x 2 - 25)= 3x(x + 5)(x - 5)The GCF of 3x 3 and 75x is 3x.Factor the difference of squares.Chapter 9 Study Guide and Review 517

Exercises Factor each polynomial, if possibl e. If the polynomial cannot befactored, write prime. See Examples 1-4 on page 502.51. 9b50. 2!/3 - 128Ya ~. ~ ~ .- ~) ..... . Check vo~ solutions. SeeSolve each equatxon oy tacto~t~53. b 2 - 16 0See pages i508-514.Concept Summa~/® If a tr~omial can be written ~ the form a 2 + 2ab + b 2 or a 2 - 2ab + b 2,t~en it can be factored as (a + b) 2 or as (a - b) 2, respectively.,For a tr~omial to be factorable as a perfect square, the first term must bea perfect square, t~e ~ddle term must be ~ice the product of the square" t and last terms, and t~e last term must be a perfect square.roots of the firs ~ _ ~ :~ ~2 = n, then x = ~.~ Square Root Proper~: For any number n ~ v, ~ ~1Determine whether 9x 2 + 2~y + 16V 2 is a perfect square trinomial.If so, factor it.Yes, 9x 2 = (3x) 2’~ Is the first term a perfect square?Yes, !6y 2 = (4Y) 2"~ Is the last term a perfect square?~ Is the middle term equal to 2(3x)(4y)? Yes, 24xy = 2(3x)(4y).9x 2 + 24xy + 16y 2 = ( 3x)2 + 2(3x)(4y) + (4Y) 2 Write as ~ + 2oh + bk= (3x + 4y) 2 FaVor using the pa~em.Solve (x - 4) 2 = 121.(X -- 4) 2 = 121x-4 = +_11x =4+-iix=4+ll or x= 15 = -7Original equationSquare Root Pyoperty121 = 11 ¯ 11Add 4 to each side.Separate into two equations.The solution set is {-7, 15}.Exercises Factor each polynomial, if possible. If the polynomial cannOt beSee Example 2 page ~lO.factored, write prime.56. a 2 + 18a + 81 ~ -K--°~’~ 9-- 57. 9k 2 - !2k +4Solve each equation. Check your solutions- See Examples 3 a~d 4 on~bL ~"~~~0" 6b 3 - 24b 2 + 24b = 0 61. 49m 2 - 126m + 81 = 062 (c-9) 2= 144 C~ {~F-J~63" 1~¯ .......... ........b2=36