Section 1.6: Factoring Sums and Differences of Cubes
#1-42: Completely factor the binomials, remember to factor out the GCF first when applicable
(if a problem is prime say so).
1) x3 + 8
signs will be ( + )( - + )
xxx 222
one of each, then pair and multiply
(x + 2)(xx – x2 + 2*2)
Solution: (x + 2)(x2 – 2x + 4)
2) x3 – 8
signs will be ( - )( + + )
xxx 222
one of each, then pair and multiply
(x - 2)(xx +x2 + 2*2)
Solution: (x - 2)(x2 + 2x + 4)
5) b3 + 27
signs will be ( + )( - + )
bbb 333
one of each, then pair and multiply
(b + 3)(bb – b3 + 3*3)
Solution: (b + 3)(b2 – 3b + 9)
7) b3 – 27
signs will be ( - )( + + )
bbb 333
one of each, then pair and multiply
(b - 3)(bb + b3 + 3*3)
Solution: (b - 3)(b2 + 3b + 9)
9) x3 + 64
signs will be ( + )( - + )
xxx 444
one of each, then pair and multiply
(x + 4)(xx – x4 + 4*4)
Solution: (x + 4)(x2 – 4x + 16)
11) x3 – 64
signs will be ( - )( + + )
xxx 444
one of each, then pair and multiply
(x - 4)(xx +x4 + 4*4)
Solution: (x - 4)(x2 + 4x + 16)
13) 8x3 – 27
signs will be ( - )( + + )
2x2x2x 333
one of each, then pair and multiply
(2x - 3)(2x2x +2x3 + 3*3)
Solution: (2x - 3)(4x2 + 6x + 9)
15) 8x3 + 27
signs will be ( + )( - + )
2x2x2x 333
one of each, then pair and multiply
(2x +3)(2x2x - 2x3 + 3*3)
Solution: (2x + 3)(4x2 - 6x + 9)
17) 27x3 – 125
signs will be ( - )( + + )
3x3x3x 555
one of each, then pair and multiply
(3x - 5)(3x3x +3x5 + 5*5)
Solution: (3x - 5)(9x2 + 15x + 25)
19) 64x3 – y3
signs will be ( - )( + + )
4x4x4x yyy
one of each, then pair and multiply
(4x - y)(4x4x +4xy + yy)
Solution: (4x - y)(16x2 + 4xy + y2)
21) x6 – y3
signs will be ( - )( + + )
x2x2x2 yyy
one of each, then pair and multiply
(x2 - y)(x2x2 +x2y + yy)
Solution: (x2 - y)(x4 + x2y + y2)
23) 27x6 – 1
signs will be ( - )( + + )
3x2 3x2 3x2 111
one of each, then pair and multiply
(3x2 - 1)(3x2 3x2 +3x21 + 1*1)
Solution: (3x2 - 1)(9x4 + 3x2+ 1)
25) 125x9 – y6
signs will be ( - )( + + )
5x3 5x3 5x3 y2 y2 y2
one of each, then pair and multiply
(5x3 – y2)(5x3 5x3 +5x3y2 + y2y2)
Solution: (5x3 – y2)(25x6 + 5x3y2+ y4)
27) 16x3 – 54
First factor out GCF of 2
= 2(8x3 – 27) Then factor what’s left inside the parenthesis.
signs will be ( - )( + + )
2x2x2x 333
one of each, then pair and multiply
2(2x - 3)(2x2x +2x3 + 3*3)
Solution: 2(2x - 3)(4x2 + 6x + 9)
29) 3x3 + 24
First factor out GCF of 3
= 3(x3 + 8) Then factor what’s left inside the parenthesis.
signs will be ( + )( - + )
xxx 222
one of each, then pair and multiply
3(x + 2)(xx – x2 + 2*2)
Solution: 3(x + 2)(x2 – 2x + 4)
31) x4 – 8x
First factor out GCF of x
= x(x3 – 8) The factor what’s left inside the parenthesis.
signs will be ( - )( + + )
xxx 222
one of each, then pair and multiply
x(x - 2)(xx +x2 + 2*2)
Solution: x(x – 2)(x2 + 2x + 4)
33) 6x4 – 48x
First factor out the GCF of 6x
= 6x(x3 – 8) then factor what’s left inside the parenthesis
signs will be ( - )( + + )
xxx 222
one of each, then pair and multiply
6x(x - 2)(xx +x2 + 2*2)
Solution: 6x(x – 2)(x2 + 2x + 4)
35) 8x5 + 125x2
First factor out the GCF of x2
= x2(8x3 + 125) then factor what’s left inside the parenthesis.
2x 2x 2x 5 5 5 (one of each, then pair and multiply)
Solution: x2(2x + 5)(4x2 - 10x + 25)
37) 27 – x3
I will leave this in the order that it is written. If I wrote the –x3 first I would get an answer that
looks different, but would be equal to my answer.
Signs will be ( - )( + + )
333 xxx
= (3 – x)(3*3 + 3x + xx)
Solution: (3 – x)(9 + 3x + x2)
39) 27 + 64x3
Again, I will leave the problem in the order it is written. My answer would look slightly
different, but would still be correct if I took the time to rewrite the problem as 64x 3 + 27
Signs ( + )( - + )
3 3 3 4x 4x 4x
=(3 + 4x)(3*3 – 3*4x + 4x*4x)
Solution: (3 + 4x)(9 – 12x + 16x2)
41) 8 + y6
Signs ( + )( - + )
2 2 2 y2 y2 y2
=(2 + y2)(2*2 – 2y2 + y2y2)
Solution: (2 + y2)(4 – 2y2 + y4)