Section 10.6 Factoring Strategies
ΒΆSubsection 10.6.1 Factoring Strategies
permalinkDeciding 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.
permalinkUsing the decision tree can guide us when we are given an expression to factor.
Example 10.6.3.
Factor the expression 4k2+12kβ404k2+12kβ40 completely.
Start by noting that the GCF is \(4\text{.}\) Factoring this out, we get
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 \(3\text{.}\) We find that \(-2\) and \(5\) work. So, the full factorization is:
Example 10.6.4.
Factor the expression 64d2+144d+8164d2+144d+81 completely.
Start by noting that the GCF is \(1\text{,}\) 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 \(A^2+2AB+B^2=(A+B)^2\text{.}\) To find \(A\) and \(B\text{,}\) take the square roots of the first and last terms, so \(A=8d\) and \(B=9\text{.}\) 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
Example 10.6.5.
Factor the expression 10x2yβ12xy2 completely.
Start by noting that the GCF is \(2xy\text{.}\) Factoring this out, we get
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 \(2xy(5x-6y)\) is the complete factorization.
Example 10.6.6.
Factor the expression 9b2β25y2 completely.
Start by noting that the GCF is \(1\text{,}\) 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, \(A^2-B^2=(A-B)(A+B)\text{.}\) To find the values for \(A\) and \(B\) that fit the patterns, just take the square roots. So \(A=3b\) since \((3b)^2=9b^2\) and \(B=5y\) since \((5y)^2=25y^2\text{.}\) So, the expression must factor as
Example 10.6.7.
Factor the expression 24w3+6w2β9w completely.
Start by noting that the GCF is \(3w\text{.}\) Factoring this out, we get
Following the decision tree, we now have a trinomial inside the parentheses where \(a\neq1\text{.}\) 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 \(2\text{.}\) The numbers \(6\) and \(-4\) work in this case. The rest of the factoring process is:
Example 10.6.8.
Factor the expression β6xy+9y+2xβ3 completely.
Start by noting that the GCF is \(1\text{,}\) 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:
Note that the negative sign in front of the \(3y\) can be factored out if you wish. That would look like:
\begin{align*} \amp=-(2x-3)(3y-1) \end{align*}Example 10.6.9.
Factor the expression 4w3β20w2+24w completely.
Start by noting that the GCF is \(4w\text{.}\) Factoring this out, we get
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 \(-5\text{.}\) Since \(-3\) and \(-2\) fit the requirements, the full factorization is:
Example 10.6.10.
Factor the expression 9β24y+16y2 completely.
Start by noting that the GCF is \(1\text{,}\) and there is no GCF to factor out. Continue along the decision tree. We now have a trinomial where both the first term, \(9\text{,}\) and last term, \(16y^2\text{,}\) look like perfect squares. To use the perfect squares difference pattern, \(A^2-2AB+B^2=(A-B)^2\text{,}\) recall that we need to mentally take the square roots of these two terms to find \(A\) and \(B\text{.}\) So, \(A=3\) since \(3^2=9\text{,}\) and \(B=4y\) since \((4y)^2=16y^2\text{.}\) Now we have to check that \(2AB\) matches \(24y\text{:}\)
So the full factorization is:
Example 10.6.11.
Factor the expression 9β25y+16y2 completely.
Start by noting that the GCF is \(1\text{,}\) 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 \(2AB=24y\text{.}\) Since our middle term is \(25y\text{,}\) 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 \(-25\text{.}\) After a brief search, we conclude that those values are \(-16\) and \(-9\text{.}\) The remainder of the factorization is:
Example 10.6.12.
Factor the expression 20x4+13x3β21x2 completely.
Start by noting that the GCF is \(x^2\text{.}\) Factoring this out, we get
Following the decision tree, we now have a trinomial inside the parentheses where \(a\neq1\) and we should try the AC method. In this case, \(ac=-420\) and we need factors of \(-420\) that add to \(13\text{.}\)
Factor Pair | Sum |
\(1\cdot-420\) | \(-419\) |
\(2\cdot-210\) | \(-208\) |
\(3\cdot-140\) | \(-137\) |
\(4\cdot-105\) | \(-101\) |
Factor Pair | Sum |
\(5\cdot-84\) | \(-79\) |
\(6\cdot-70\) | \(-64\) |
\(7\cdot-60\) | \(-53\) |
\(10\cdot-42\) | \(-32\) |
Factor Pair | Sum |
\(12\cdot-35\) | \(-23\) |
\(14\cdot-30\) | \(-16\) |
\(15\cdot-28\) | \(-13\) |
\(20\cdot-21\) | \(-1\) |
In the table of the factor pairs of \(-420\) we find \(15+(-28)=-13\text{,}\) the opposite of what we want, so we want the opposite numbers: \(-15\) and \(28\text{.}\) The rest of the factoring process is shown:
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
64B2β216Bb+140b2
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
6m6+384m3x3
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
49n3β35n2β28n+20
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
4q4β2916q
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
9y2β24y+16
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
3r4β39r3+120r2
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
40x2β408xa+432a2
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
b3βb2β2b+2
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
54A6β686A3B3
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
63nβ81
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
32m3β50mp2
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
n3+64A3
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
210q3tβ168q3β270q2t+216q2
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
4y5β2916y2p3
Factoring
15.
Factor the given polynomial.
6x+6=
16.
Factor the given polynomial.
β3yβ3=
17.
Factor the given polynomial.
36y2+27=
18.
Factor the given polynomial.
30r4β12r3+42r2=
19.
Factor the given polynomial.
6r+12r2+15r3=
20.
Factor the given polynomial.
8xy+8y=
21.
Factor the given polynomial.
54x5y8β6x4y8+42x3y8=
22.
Factor the given polynomial.
t2β2t+3tβ6=
23.
Factor the given polynomial.
xy+2x+8y+16=
24.
Factor the given polynomial.
x3β3+2x3yβ6y=
25.
Factor the given polynomial.
y2β3yβ4=
26.
Factor the given polynomial.
5y2β2yβ7=
27.
Factor the given polynomial.
2r2y2+3ryβ9=
28.
Factor the given polynomial.
2r2β6r+5=
29.
Factor the given polynomial.
4r2β7rβ2=
30.
Factor the given polynomial.
8t2+22t+15=
31.
Factor the given polynomial.
12t2β23t+10=
32.
Factor the given polynomial.
3x2+10xr+3r2=
33.
Factor the given polynomial.
3x2β13xy+12y2=
34.
Factor the given polynomial.
4y2+3ytβ7t2=
35.
Factor the given polynomial.
8y2+22yr+9r2=
36.
Factor the given polynomial.
8r2β18rx+9x2=
37.
Factor the given polynomial.
12r2β8rβ4=
38.
Factor the given polynomial.
15r2t2β3rtβ12=
39.
Factor the given polynomial.
10t9+25t8+15t7=
40.
Factor the given polynomial.
6t10β9t9+3t8=
41.
Factor the given polynomial.
6x2+20xy+14y2=
42.
Factor the given polynomial.
10x2β34xy+12y2=
43.
Factor the given polynomial.
y2+9y+8=
44.
Factor the given polynomial.
y2β6y+5=
45.
Factor the given polynomial.
r2+10rx+16x2=
46.
Factor the given polynomial.
r2y2β4ryβ12=
47.
Factor the given polynomial.
r2β10ry+24y2=
48.
Factor the given polynomial.
4t2x2+12tx+8=
49.
Factor the given polynomial.
2t2+8tβ10=
50.
Factor the given polynomial.
3x4+18x3+24x2=
51.
Factor the given polynomial.
7x9β28x8+21x7=
52.
Factor the given polynomial.
2x2y+6xy+4y=
53.
Factor the given polynomial.
2x2yβ18xy+28y=
54.
Factor the given polynomial.
2x2y3β10xy2+8y=
55.
Factor the given polynomial.
x2y2+6x2yzβ7x2z2=
56.
Factor the given polynomial.
r2+0.9r+0.2=
57.
Factor the given polynomial.
t2β144=
58.
Factor the given polynomial.
t2r2β16=
59.
Factor the given polynomial.
9βx2=
60.
Factor the given polynomial.
x4β121=
61.
Factor the given polynomial.
y12β49=
62.
Factor the given polynomial.
x6β36y14=
63.
Factor the given polynomial.
81y4β16=
64.
Factor the given polynomial.
3r3β147r=
65.
Factor the given polynomial.
r2+4=
66.
Factor the given polynomial.
32β8t2=
67.
Factor the given polynomial.
t2+12t+36=
68.
Factor the given polynomial.
x2β12xy+36y2=
69.
Factor the given polynomial.
x2β18x+81=
70.
Factor the given polynomial.
36y2β12y+1=
71.
Factor the given polynomial.
y2+18yx+81x2=
72.
Factor the given polynomial.
9y2+30yr+25r2=
73.
Factor the given polynomial.
98r2y2+28ry+2=
74.
Factor the given polynomial.
4r10+4r9+r8=
75.
Factor the given polynomial.
98t8+28t7+2t6=
76.
Factor the given polynomial.
0.16tβt3=
77.
Factor the given polynomial.
2x4β162=
78.
Factor the given polynomial.
x2β14x+49β64y2=