Learning Objectives
- Recognize and use the appropriate method to factor a polynomial completely
Recognize and Use the Appropriate Method to Factor a Polynomial Completely
You have now become acquainted with all the methods of factoring that you will need in this course. The following chart summarizes all the factoring methods we have covered, and outlines a strategy you should use when factoring polynomials.
General Strategy for Factoring Polynomials
How To
Use a general strategy for factoring polynomials.
- Step 1.
Is there a greatest common factor?
Factor it out. - Step 2.
Is the polynomial a binomial, trinomial, or are there more than three terms?
If it is a binomial:- Is it a sum?
Of squares? Sums of squares do not factor.
Of cubes? Use the sum of cubes pattern. - Is it a difference?
Of squares? Factor as the product of conjugates.
Of cubes? Use the difference of cubes pattern.
- Is it of the form x2+bx+c? Undo FOIL.
- Is it of the form ax2+bx+c?
If a and c are squares, check if it fits the trinomial square pattern.
Use the trial and error or “ac” method.
- Use the grouping method.
- Is it a sum?
- Step 3.
Check.
Is it factored completely?
Do the factors multiply back to the original polynomial?
Remember, a polynomial is completely factored if, other than monomials, its factors are prime!
Example 6.35
Factor completely: 7x3−21x2−70x.
Solution
7x3−21x2−70xIs there a GCF? Yes,7x.Factor out the GCF.7x(x2−3x−10)In the parentheses, is it a binomial, trinomial,or are there more terms?Trinomial with leading coefficient 1.“Undo” FOIL.7x(x)(x)7x(x+2)(x−5)Is the expression factored completely? Yes.Neither binomial can be factored.Check your answer.Multiply.7x(x+2)(x−5)7x(x2−5x+2x−10)7x(x2−3x−10)7x3−21x2−70x✓
Try It 6.69
Factor completely: 8y3+16y2−24y.
Try It 6.70
Factor completely: 5y3−15y2−270y.
Be careful when you are asked to factor a binomial as there are several options!
Example 6.36
Factor completely: 24y2−150.
Solution
24y2−150Is there a GCF? Yes, 6.Factor out the GCF.6(4y2−25)In the parentheses, is it a binomial, trinomialor are there more than three terms? Binomial.Is it a sum? No.Is it a difference? Of squares or cubes? Yes, squares.6((2y)2−(5)2)Write as a product of conjugates.6(2y−5)(2y+5)Is the expression factored completely?Neither binomial can be factored.Check:Multiply.6(2y−5)(2y+5)6(4y2−25)24y2−150✓
Try It 6.71
Factor completely: 16x3−36x.
Try It 6.72
Factor completely: 27y2−48.
The next example can be factored using several methods. Recognizing the trinomial squares pattern will make your work easier.
Example 6.37
Factor completely: 4a2−12ab+9b2.
Solution
4a2−12ab+9b2Is there a GCF? No.Is it a binomial, trinomial, or are there more terms?Trinomial witha≠1.But the first term is a perfect square.Is the last term a perfect square? Yes.(2a)2−12ab+(3b)2Does it fit the pattern,a2−2ab+b2?Yes.(2a)2↘−12ab+−2(2a)(3b)↙(3b)2Write it as a square.(2a−3b)2Is the expression factored completely? Yes.The binomial cannot be factored.Check your answer.Multiply.(2a−3b)2(2a)2−2·2a·3b+(3b)24a2−12ab+9b2✓
Try It 6.73
Factor completely: 4x2+20xy+25y2.
Try It 6.74
Factor completely: 9x2−24xy+16y2.
Remember, sums of squares do not factor, but sums of cubes do!
Example 6.38
Factor completely 12x3y2+75xy2.
Solution
12x3y2+75xy2Is there a GCF? Yes,3xy2.Factor out the GCF.3xy2(4x2+25)In the parentheses, is it a binomial, trinomial, or arethere more than three terms? Binomial.Is it a sum? Of squares? Yes.Sums of squares are prime.Is the expression factored completely? Yes.Check:Multiply.3xy2(4x2+25)12x3y2+75xy2✓
Try It 6.75
Factor completely: 50x3y+72xy.
Try It 6.76
Factor completely: 27xy3+48xy.
When using the sum or difference of cubes pattern, being careful with the signs.
Example 6.39
Factor completely: 24x3+81y3.
Solution
Is there a GCF? Yes, 3. | |
Factor it out. | |
In the parentheses, is it a binomial, trinomial, of are there more than three terms? Binomial. |
|
Is it a sum or difference? Sum. | |
Of squares or cubes? Sum of cubes. | |
Write it using the sum of cubes pattern. | |
Is the expression factored completely? Yes. | |
Check by multiplying. |
Try It 6.77
Factor completely: 250m3+432n3.
Try It 6.78
Factor completely: 2p3+54q3.
Example 6.40
Factor completely: 3x5y−48xy.
Solution
3x5y−48xyIs there a GCF? Factor out3xy3xy(x4−16)Is the binomial a sum or difference? Of squares or cubes?Write it as a difference of squares.3xy((x2)2−(4)2)Factor it as a product of conjugates3xy(x2−4)(x2+4)The first binomial is again a difference of squares.3xy((x)2−(2)2)(x2+4)Factor it as a product of conjugates.3xy(x−2)(x+2)(x2+4)Is the expression factored completely? Yes.Check your answer.Multiply.3xy(x−2)(x+2)(x2+4)3xy(x2−4)(x2+4)3xy(x4−16)3x5y−48xy✓
Try It 6.79
Factor completely: 4a5b−64ab.
Try It 6.80
Factor completely: 7xy5−7xy.
Example 6.41
Factor completely: 4x2+8bx−4ax−8ab.
Solution
4x2+8bx−4ax−8abIs there a GCF? Factor out the GCF, 4.4(x2+2bx−ax−2ab)There are four terms. Use grouping.4[x(x+2b)−a(x+2b)]4(x+2b)(x−a)Is the expression factored completely? Yes.Check your answer.Multiply.4(x+2b)(x−a)4(x2−ax+2bx−2ab)4x2+8bx−4ax−8ab✓
Try It 6.81
Factor completely: 6x2−12xc+6bx−12bc.
Try It 6.82
Factor completely: 16x2+24xy−4x−6y.
Taking out the complete GCF in the first step will always make your work easier.
Example 6.42
Factor completely: 40x2y+44xy−24y.
Solution
40x2y+44xy−24yIs there a GCF? Factor out the GCF,4y.4y(10x2+11x−6)Factor the trinomial witha≠1.4y(10x2+11x−6)4y(5x−2)(2x+3)Is the expression factored completely? Yes.Check your answer.Multiply.4y(5x−2)(2x+3)4y(10x2+11x−6)40x2y+44xy−24y✓
Try It 6.83
Factor completely: 4p2q−16pq+12q.
Try It 6.84
Factor completely: 6pq2−9pq−6p.
When we have factored a polynomial with four terms, most often we separated it into two groups of two terms. Remember that we can also separate it into a trinomial and then one term.
Example 6.43
Factor completely: 9x2−12xy+4y2−49.
Solution
9x2−12xy+4y2−49Is there a GCF? No.With more than 3 terms, use grouping. Last 2 termshave no GCF. Try grouping first 3 terms.9x2−12xy+4y2−49Factor the trinomial witha≠1.But the first term is aperfect square.Is the last term of the trinomial a perfect square? Yes.(3x)2−12xy+(2y)2−49Does the trinomial fit the pattern,a2−2ab+b2?Yes.(3x)2↘−12xy+−2(3x)(2y)↙(2y)2−49Write the trinomial as a square.(3x−2y)2−49Is this binomial a sum or difference? Of squares orcubes? Write it as a difference of squares.(3x−2y)2−72Write it as a product of conjugates.((3x−2y)−7)((3x−2y)+7)(3x−2y−7)(3x−2y+7)Is the expression factored completely? Yes.Check your answer.Multiply.(3x−2y−7)(3x−2y+7)9x2−6xy−21x−6xy+4y2+14y+21x−14y−499x2−12xy+4y2−49✓
Try It 6.85
Factor completely: 4x2−12xy+9y2−25.
Try It 6.86
Factor completely: 16x2−24xy+9y2−64.
Section 6.4 Exercises
Practice Makes Perfect
Recognize and Use the Appropriate Method to Factor a Polynomial Completely
In the following exercises, factor completely.
8x2−9x−3
75m3+12m
49b2−36a2
36q2−100
49b2−112b+64
64x2+16xy+y2
30n2+30n+72
4x5y−32x2y
m4−81
48x5y2−243xy2
12ab−6a+10b−5
5q2−15q−90
5m4n+320mn4
25x2+35xy+49y2
3v4−768
60x2y−75xy+30y
64x3+125y3
y6+1
16x2−24xy+9y2−64
(4x−5)2−7(4x−5)+12
Writing Exercises
The difference of squares y4−625 can be factored as (y2−25)(y2+25). But it is not completely factored. What more must be done to completely factor.
Of all the factoring methods covered in this chapter (GCF, grouping, undo FOIL, ‘ac’ method, special products) which is the easiest for you? Which is the hardest? Explain your answers.
Create three factoring problems that would be good test questions to measure your knowledge of factoring. Show the solutions.
Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?