Section 1.7: A Review of all the Factoring Strategies – Mixed Up
#1-44: Factor completely, state if a polynomial is prime.
1) a2 + 16
This is a sum of squares so it is prime.
Solution: Prime
2) 81y2 – 4
This is a difference of squares. I take the square root of the numbers, and divide the exponent
by 2. Each parenthesis is the same, other than the signs.
Solution: (9y + 2)(9y – 2)
5) b3 + 64
This is a sum of cubes
Signs ( + )( - + )
b b b 4 4 4 (first parenthesis one of each, second pair and multiply)
= (b + 4)(bb – b4 + 4*4)
Solution: (b + 4)(b2 – 4b + 16)
7) 64x3 – 1
This is a difference of cubes.
Signs will be ( - )( + + )
4x 4x 4x 1 1 1 (first parenthesis one of each, second pair and multiply)
= (4x – 1)(4x*4x + 4x*1 + 1*1)
Solution: (4x – 1)(16x2 + 4x + 1)
9) 2x2 – 3x – 9
This is a bottoms up problem.
Multiply first x2 – 3x – 18
now factor this (x + 3)(x – 6)
now divide by 2, reduce and bottoms up.
3
6
(𝑥 + 2) (𝑥 − 2)
Solution: (2x + 3)(x – 3)
11) -4x2 + 6x + 18
First factor out the GCF of -2
= -2(2x2 – 3x – 9)
Now bottoms up what is left inside the parenthesis.
-2(x2 – 3x - 18)
-2(x+3)(x-6)
3
6
−2 (𝑥 + 2) (𝑥 − 2)
Solution: -2(2x + 3)(x – 3)
13) 3x2 – 13x + 10
This is bottoms up.
x2 - 13x + 30
(x – 10)(x – 3)
(𝑥 −
10
3
) (𝑥 − 3)
3
Solution: (10x – 3)(x – 1)
15) -w2 + 8w – 15
Factor out a -1
= -1(w2 – 8w + 15)
Then factor what’s left. My lasts need to add to -8 and multiply to 15 the signs are ( - )( - )
Solution: -1(w – 3)(w – 5) also can write -(w – 3)(w – 5)
17) x2 – 2x + 15
Lasts need to multiply to 15 and add to -2. No such numbers exist, hint signs have to be ( - )(- )
it looks like (x + 3)(x – 5) works, but it doesn’t as this FOILS to x2 – 2x – 15 and has the wrong
sign on the 15.
Solution: Prime
19) 5x2 + 10x + 6x + 12
This has 4 terms, so try grouping.
= (5x2 + 10x) + (6x + 12)
= 5x(x + 2) + 6(x + 2)
Solution: (x + 2)(5x + 6)
21) x2 + 5x + 9
Lasts need to multiply to 9 and add to 5. No such numbers exist, so this is prime.
Solution: Prime
23) 6x4 – 6x
First factor out the GCF of 6x
= 6x(x3 – 1) now factor the cube problem that is left inside the parenthesis.
Signs 6x( - )( + + )
xxx 111
= 6x(x – 1)(xx + x1 + 1*1)
Solution: 6x(x – 1)( x2 + x + 1)
25) 2x2 – 8
First factor out a GCF of 2
= 2(x2 – 4)
Now factor what’s left inside the parenthesis.
Solution: 2(x + 2)(x – 2)
27) 3x2 + 12
There is a common factor of 3. Once you factor it out you need to see if what is left inside the
parenthesis factors.
= 3(x2 + 4)
What is left inside the parenthesis is a sum of squares and it doesn’t factor.
Solution: 3(x2 + 4)
29) 3x2 – 5x – 6x + 10
This has 4 terms so group.
= (3x2 – 5x) + ((-6x)+10)
= x(3x – 5) + -2(3x – 5)
= x(3x – 5) – 2(3x – 5)
Solution: (3x – 5)(x – 2)
31) x2 + x + 24x + 24
This is grouping
= (x2 + x) + (24x + 24)
= x(x + 1) + 24(x + 1)
Solution: (x + 1)(x + 24)
33) 6x2 + 13x + 6
This is bottoms up
x2 + 13x + 36
(x + 4)(x + 9)
4
9
2
3
(𝑥 + 6) (𝑥 + 6)
(𝑥 + 3) (𝑥 + 2)
Solution: (3x + 2)(2x + 3)
35) -x2 + 5x + 6
First factor out a -1
= -1(x2 – 5x – 6)
Now factor what’s left inside the parenthesis.
Solution: -1(x + 1)(x – 6)
37) -3x2 – 12x +36
First factor out the GCF or -3
= -3(x2 + 4x – 12)
Now factor what’s left inside the parenthesis. The signs are ( + )( -) and the lasts need to
multiply to -12 and add to 4. They will be +6 and -2
Solution: -3(x + 6)(x – 2)
39) z2 – 5z + 4
Lasts have to multiply to 4 and add to -5.
I also know my signs are ( - )( - )
My lasts must be (-1)(-4) since these numbers multiply to 4 and add to -5
Solution: (z – 1)(z – 4)
41) x2 – 14x – 15
Lasts need to multiply to -15 and add to -14.
I also know my signs are ( + )( - )
Lasts must be + 1 and – 15
Solution: (x + 1)(x – 15)
43) a2 – a – 2
Lasts have to multiply to -2 and add to -1.
I also know the signs are ( + )( - )
Lasts need to be +1 – 2
Solution: (a + 1)(a – 2)