320 MCGRAW-HILL’S SAT
Concept Review 5
To write it as a product (result of multiplication).
x^2 −b^2 =(x+b)(x−b)
x^2 + 2 xb+b^2 =(x+b)(x+b)
x^2 − 2 xb+b^2 =(x−b)(x−b)
x^2 +(a+b)x+ab=(x+a)(x+b)
If the product of a set of numbers is 0, then at
least one of the numbers must be 0.
108 =(2)(2)(3)(3)(3)
21mn=(3)(7)(m)(n) and 75n^2 =(3)(5)(5)(n)(n), so
the least common multiple is
(3)(5)(5)(7)(m)(n)(n) = 525 mn^2.
108x^6 =(2)(2)(3)(3)(3)(x)(x)(x)(x)(x)(x) and 90x^4 =
(2)(3)(3)(5)(x)(x)(x)(x), so the greatest common
factor is (2)(3)(x)(x)(x)(x) = 6 x^4.
1 − 49 x^4 =(1 − 7 x^2 )(1 + 7 x^2 )
8.m^2 + 7 m+ 12 =(m+4)(m+3)
16x^2 − 40 x+ 25 =(4x−5)(4x−5) =(4x−5)^2
()yy yyy+ (^33) ()− =−+−=−^2 333 3y
(^22)
11.
12.
- 4 x^2 = 12 x
Subtract 12x: 4 x^2 − 12 x= 0
Factor: 4 x(x−3) = 0
Use zero product property: x=0 or 3 - x^2 − 8 x= 33
Subtract 33: x^2 − 8 x− 33 = 0
Factor: (x−11)(x+3) = 0
Use zero product property: x=11 or − 3 - 3 xz− 3 yz= 60
Factor: 3 z(x−y) = 60
Substitute z=5: 15(x−y) = 60
Divide by 15: (x−y) = 4
=− 912520 xx^2 +325
3 3 3 25 3 25 2525
2
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Answer Key 5: Factoring
SAT Practice 5
1.C 72 =(2)(2)(2)(3)(3) and 54 =(2)(3)(3)(3), so the
least common multiple is (2)(2)(2)(3)(3)(3) =216.
216 minutes is 3 hours 36 minutes.
2.E You can solve this one simply by plugging in
x=7 and y=1 and evaluating (7 −1)^2 −(7 +1)^2 = 36 −
64 =−28. Or you could do the algebra: (x−y)^2 −(x+y)^2
FOIL: (x^2 − 2 xy+y^2 ) −(x^2 + 2 xy+y^2 )
Simplify: − 4 xy
Substitute xy=7: −4(7) =− 28
- 5 (x+a)(x+1) =x^2 + 6 x+a
FOIL: x^2 +x+ax+a=x^2 + 6 x+a
Subtract x^2 and a: x+ax= 6 x
Factor: x(1 +a) = 6 x
Divide by x: 1 +a= 6
Subtract 1: a= 5
4.A The slope is “the rise over the run,” which is
the difference of the y’s divided by the difference of
the x’s:Or you can just choose values for mand n,like 2 and 1,
and evaluate the slope numerically. The slope between
(1, 1) and (2, 4) is 3, and the expression in (A) is the only
one that gives a value of 3.
5.A (a+b)^2 =(a+b)(a+b) =a^2 + 2 ab+b^2
Commute: =a^2 +b^2 + 2 ab
Substitute ab=− 2
and a^2 +b^2 =8: =(8) +2(−2)
= 4
6.D Factor: f^2 −g^2 =(f+g)(f−g)
Substitute f^2 −g^2 =− 10
and f+g=2: − 10 =2(f−g)
Divide by 2: − 5 =f−gmn
mnmnmn
mnmn(^22) −
−