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Section 10.6 Factoring Strategies

Figure 10.6.1. Alternative Video Lesson

Subsection 10.6.1 Factoring Strategies

Deciding which method to use when factoring a random polynomial can seem like a daunting task. Understanding all of the techniques that we have learned and how they fit together can be done using a decision tree.
A decision tree to help factoring.
Figure 10.6.2. Factoring Decision Tree
Using the decision tree can guide us when we are given an expression to factor.

Example 10.6.3.

Factor the expression 4k2+12k40 completely.
Explanation.
Start by noting that the GCF is 4. Factoring this out, we get
4k2+12k40=4(k2+3k10).
Following the decision tree, we now have a trinomial where the leading coefficient is 1 and we need to look for factors of 10 that add to 3. We find that 2 and 5 work. So, the full factorization is:
4k2+12k40=4(k2+3k10)=4(k2)(k+5)

Example 10.6.4.

Factor the expression 64d2+144d+81 completely.
Explanation.
Start by noting that the GCF is 1, and there is no GCF to factor out. We continue along the decision tree for a trinomial. Notice that both 64 and 81 are perfect squares and that this expression might factor using the pattern A2+2AB+B2=(A+B)2. To find A and B, take the square roots of the first and last terms, so A=8d and B=9. We have to check that the middle term is correct: since 2AB=2(8d)(9)=144d matches our middle term, the expression must factor as
64d2+144d+81=(8d+9)2.

Example 10.6.5.

Factor the expression 10x2y12xy2 completely.
Explanation.
Start by noting that the GCF is 2xy. Factoring this out, we get
10x2y12xy2=2xy(5x6y).
Since we have a binomial inside the parentheses, the only options on the decision tree for a binomial involve squares or cubes. Since there are none, we conclude that 2xy(5x6y) is the complete factorization.

Example 10.6.6.

Factor the expression 9b225y2 completely.
Explanation.
Start by noting that the GCF is 1, and there is no GCF to factor out. We continue along the decision tree for a binomial and notice that we now have a difference of squares, A2B2=(AB)(A+B). To find the values for A and B that fit the patterns, just take the square roots. So A=3b since (3b)2=9b2 and B=5y since (5y)2=25y2. So, the expression must factor as
9b225y2=(3b5y)(3b+5y).

Example 10.6.7.

Factor the expression 24w3+6w29w completely.
Explanation.
Start by noting that the GCF is 3w. Factoring this out, we get
24w3+6w29w=3w(8w2+2w3).
Following the decision tree, we now have a trinomial inside the parentheses where a1. We should try the AC method because neither 8 nor 3 are perfect squares. In this case, ac=24 and we must find two factors of 24 that add to be 2. The numbers 6 and 4 work in this case. The rest of the factoring process is:
24w3+6w29w=3w(8w2+2w3)=3w(8w2+6w4w3)=3w((8w2+6w)+(4w3))=3w(2w(4w+3)1(4w+3))=3w(4w+3)(2w1)

Example 10.6.8.

Factor the expression 6xy+9y+2x3 completely.
Explanation.
Start by noting that the GCF is 1, and there is no GCF to factor out. We continue along the decision tree. Since we have a four-term polynomial, we should try to factor by grouping. The full process is:
6xy+9y+2x3=(6xy+9y)+(2x3)=3y(2x3)+1(2x3)=(2x3)(3y+1)
Note that the negative sign in front of the 3y can be factored out if you wish. That would look like:
=(2x3)(3y1)

Example 10.6.9.

Factor the expression 4w320w2+24w completely.
Explanation.
Start by noting that the GCF is 4w. Factoring this out, we get
4w320w2+24w=4w(w25w+6).
Following the decision tree, we now have a trinomial with a=1 inside the parentheses. So, we can look for factors of 6 that add up to 5. Since 3 and 2 fit the requirements, the full factorization is:
4w320w2+24w=4w(w25w+6)=4w(w3)(w2)

Example 10.6.10.

Factor the expression 924y+16y2 completely.
Explanation.
Start by noting that the GCF is 1, and there is no GCF to factor out. Continue along the decision tree. We now have a trinomial where both the first term, 9, and last term, 16y2, look like perfect squares. To use the perfect squares difference pattern, A22AB+B2=(AB)2, recall that we need to mentally take the square roots of these two terms to find A and B. So, A=3 since 32=9, and B=4y since (4y)2=16y2. Now we have to check that 2AB matches 24y:
2AB=2(3)(4y)=24y.
So the full factorization is:
924y+16y2=(34y)2.

Example 10.6.11.

Factor the expression 925y+16y2 completely.
Explanation.
Start by noting that the GCF is 1, and there is no GCF to factor out. Since we now have a trinomial where both the first term and last term are perfect squares in exactly the same way as in Example 10. However, we cannot apply the perfect squares method to this problem because it worked when 2AB=24y. Since our middle term is 25y, we can be certain that it won’t be a perfect square.
Continuing on with the decision tree, our next option is to use the AC method. You might be tempted to rearrange the order of the terms, but that is unnecessary. In this case, ac=144 and we need to come up with two factors of 144 that add to be 25. After a brief search, we conclude that those values are 16 and 9. The remainder of the factorization is:
925y+16y2=916y9y+16y2=(916y)+(9y+16y2)=1(916y)y(9+16y)=(916y)(1y)

Example 10.6.12.

Factor the expression 20x4+13x321x2 completely.
Explanation.
Start by noting that the GCF is x2. Factoring this out, we get
20x4+13x321x2=x2(20x2+13x21).
Following the decision tree, we now have a trinomial inside the parentheses where a1 and we should try the AC method. In this case, ac=420 and we need factors of 420 that add to 13.
Factor Pair Sum
1420 419
2210 208
3140 137
4105 101
Factor Pair Sum
584 79
670 64
760 53
1042 32
Factor Pair Sum
1235 23
1430 16
1528 13
2021 1
In the table of the factor pairs of 420 we find 15+(28)=13, the opposite of what we want, so we want the opposite numbers: 15 and 28. The rest of the factoring process is shown:
20x4+13x321x2=x2(20x2+13x21)=x2(20x215x+28x21)=x2((20x215x)+(28x21))=x2(5x(4x3)+7(4x3))=x2(4x3)(5x+7)

Reading Questions 10.6.2 Reading Questions

1.

Do you find a factoring flowchart helpful?

Exercises 10.6.3 Exercises

Strategies.

1.
Which factoring techniques/tools will be useful for factoring the polynomial below? Check all that apply.
  • Factoring out a GCF
  • Factoring by grouping
  • Finding two numbers that multiply to the constant term and sum to the linear coefficient
  • The AC Method
  • Difference of Squares
  • Difference of Cubes
  • Sum of Cubes
  • Perfect Square Trinomial
  • None of the above
30y229y+4
2.
Which factoring techniques/tools will be useful for factoring the polynomial below? Check all that apply.
  • Factoring out a GCF
  • Factoring by grouping
  • Finding two numbers that multiply to the constant term and sum to the linear coefficient
  • The AC Method
  • Difference of Squares
  • Difference of Cubes
  • Sum of Cubes
  • Perfect Square Trinomial
  • None of the above
8t3+4096
3.
Which factoring techniques/tools will be useful for factoring the polynomial below? Check all that apply.
  • Factoring out a GCF
  • Factoring by grouping
  • Finding two numbers that multiply to the constant term and sum to the linear coefficient
  • The AC Method
  • Difference of Squares
  • Difference of Cubes
  • Sum of Cubes
  • Perfect Square Trinomial
  • None of the above
x281
4.
Which factoring techniques/tools will be useful for factoring the polynomial below? Check all that apply.
  • Factoring out a GCF
  • Factoring by grouping
  • Finding two numbers that multiply to the constant term and sum to the linear coefficient
  • The AC Method
  • Difference of Squares
  • Difference of Cubes
  • Sum of Cubes
  • Perfect Square Trinomial
  • None of the above
c225
5.
Which factoring techniques/tools will be useful for factoring the polynomial below? Check all that apply.
  • Factoring out a GCF
  • Factoring by grouping
  • Finding two numbers that multiply to the constant term and sum to the linear coefficient
  • The AC Method
  • Difference of Squares
  • Difference of Cubes
  • Sum of Cubes
  • Perfect Square Trinomial
  • None of the above
147A2378AB+243B2
6.
Which factoring techniques/tools will be useful for factoring the polynomial below? Check all that apply.
  • Factoring out a GCF
  • Factoring by grouping
  • Finding two numbers that multiply to the constant term and sum to the linear coefficient
  • The AC Method
  • Difference of Squares
  • Difference of Cubes
  • Sum of Cubes
  • Perfect Square Trinomial
  • None of the above
C29Ct+8t2
7.
Which factoring techniques/tools will be useful for factoring the polynomial below? Check all that apply.
  • Factoring out a GCF
  • Factoring by grouping
  • Finding two numbers that multiply to the constant term and sum to the linear coefficient
  • The AC Method
  • Difference of Squares
  • Difference of Cubes
  • Sum of Cubes
  • Perfect Square Trinomial
  • None of the above
1125m3+72p3
8.
Which factoring techniques/tools will be useful for factoring the polynomial below? Check all that apply.
  • Factoring out a GCF
  • Factoring by grouping
  • Finding two numbers that multiply to the constant term and sum to the linear coefficient
  • The AC Method
  • Difference of Squares
  • Difference of Cubes
  • Sum of Cubes
  • Perfect Square Trinomial
  • None of the above
pA2p7A+14
9.
Which factoring techniques/tools will be useful for factoring the polynomial below? Check all that apply.
  • Factoring out a GCF
  • Factoring by grouping
  • Finding two numbers that multiply to the constant term and sum to the linear coefficient
  • The AC Method
  • Difference of Squares
  • Difference of Cubes
  • Sum of Cubes
  • Perfect Square Trinomial
  • None of the above
1458x32r3
10.
Which factoring techniques/tools will be useful for factoring the polynomial below? Check all that apply.
  • Factoring out a GCF
  • Factoring by grouping
  • Finding two numbers that multiply to the constant term and sum to the linear coefficient
  • The AC Method
  • Difference of Squares
  • Difference of Cubes
  • Sum of Cubes
  • Perfect Square Trinomial
  • None of the above
63n81
11.
Which factoring techniques/tools will be useful for factoring the polynomial below? Check all that apply.
  • Factoring out a GCF
  • Factoring by grouping
  • Finding two numbers that multiply to the constant term and sum to the linear coefficient
  • The AC Method
  • Difference of Squares
  • Difference of Cubes
  • Sum of Cubes
  • Perfect Square Trinomial
  • None of the above
30t243t+15
12.
Which factoring techniques/tools will be useful for factoring the polynomial below? Check all that apply.
  • Factoring out a GCF
  • Factoring by grouping
  • Finding two numbers that multiply to the constant term and sum to the linear coefficient
  • The AC Method
  • Difference of Squares
  • Difference of Cubes
  • Sum of Cubes
  • Perfect Square Trinomial
  • None of the above
7a3+5103
13.
Which factoring techniques/tools will be useful for factoring the polynomial below? Check all that apply.
  • Factoring out a GCF
  • Factoring by grouping
  • Finding two numbers that multiply to the constant term and sum to the linear coefficient
  • The AC Method
  • Difference of Squares
  • Difference of Cubes
  • Sum of Cubes
  • Perfect Square Trinomial
  • None of the above
40c335c232c+28
14.
Which factoring techniques/tools will be useful for factoring the polynomial below? Check all that apply.
  • Factoring out a GCF
  • Factoring by grouping
  • Finding two numbers that multiply to the constant term and sum to the linear coefficient
  • The AC Method
  • Difference of Squares
  • Difference of Cubes
  • Sum of Cubes
  • Perfect Square Trinomial
  • None of the above
5A3625

Factoring.

15.
Factor the given polynomial.
3r+3=
16.
Factor the given polynomial.
8t8=
17.
Factor the given polynomial.
72t2+40=
18.
Factor the given polynomial.
63t4+36t3+54t2=
19.
Factor the given polynomial.
72x+48x2+16x3=
20.
Factor the given polynomial.
3xy+3y=
21.
Factor the given polynomial.
90x5y4+100x4y4+60x3y4=
22.
Factor the given polynomial.
y2+8y+6y+48=
23.
Factor the given polynomial.
xy6x5y+30=
24.
Factor the given polynomial.
x3+7+9x3y+63y=
25.
Factor the given polynomial.
t2+3t70=
26.
Factor the given polynomial.
2t2t10=
27.
Factor the given polynomial.
2t2x23tx5=
28.
Factor the given polynomial.
3x2x+7=
29.
Factor the given polynomial.
4x215x4=
30.
Factor the given polynomial.
6y2+17y+5=
31.
Factor the given polynomial.
6y229y+9=
32.
Factor the given polynomial.
2r2+13rx+15x2=
33.
Factor the given polynomial.
3r211rt+6t2=
34.
Factor the given polynomial.
6t213tr15r2=
35.
Factor the given polynomial.
6t2+19tx+8x2=
36.
Factor the given polynomial.
10t217tr+3r2=
37.
Factor the given polynomial.
6x2+2x20=
38.
Factor the given polynomial.
27x2r2+18xr9=
39.
Factor the given polynomial.
20y8+30y7+10y6=
40.
Factor the given polynomial.
10y922y8+4y7=
41.
Factor the given polynomial.
4x2+22xy+18y2=
42.
Factor the given polynomial.
15x225xy+10y2=
43.
Factor the given polynomial.
t2+12t+32=
44.
Factor the given polynomial.
t23t+2=
45.
Factor the given polynomial.
t2+8tr+12r2=
46.
Factor the given polynomial.
x2y2+2xy15=
47.
Factor the given polynomial.
x27xt+12t2=
48.
Factor the given polynomial.
2y2r2+10yr+12=
49.
Factor the given polynomial.
7y221y28=
50.
Factor the given polynomial.
2r6+22r5+20r4=
51.
Factor the given polynomial.
9r1027r9+18r8=
52.
Factor the given polynomial.
3x2y+15xy+18y=
53.
Factor the given polynomial.
9x2y27xy+18y=
54.
Factor the given polynomial.
4x2y324xy2+20y=
55.
Factor the given polynomial.
x2y2+7x2yz8x2z2=
56.
Factor the given polynomial.
x2+1.1x+0.24=
57.
Factor the given polynomial.
y249=
58.
Factor the given polynomial.
y2x2144=
59.
Factor the given polynomial.
25r2=
60.
Factor the given polynomial.
r449=
61.
Factor the given polynomial.
t69=
62.
Factor the given polynomial.
x14121y12=
63.
Factor the given polynomial.
16t41=
64.
Factor the given polynomial.
7x328x=
65.
Factor the given polynomial.
x2+81=
66.
Factor the given polynomial.
1255y2=
67.
Factor the given polynomial.
y2+2y+1=
68.
Factor the given polynomial.
r220rx+100x2=
69.
Factor the given polynomial.
r210r+25=
70.
Factor the given polynomial.
144t224t+1=
71.
Factor the given polynomial.
t2+10tr+25r2=
72.
Factor the given polynomial.
25t2+30ty+9y2=
73.
Factor the given polynomial.
80x2t2+40xt+5=
74.
Factor the given polynomial.
64x8+16x7+x6=
75.
Factor the given polynomial.
32y9+16y8+2y7=
76.
Factor the given polynomial.
yy3=
77.
Factor the given polynomial.
5r480=
78.
Factor the given polynomial.
x2+6x+916y2=
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