6.4 General Strategy for Factoring Polynomials

Learning objectives

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.

Remember, a polynomial is completely factored if, other than monomials, its factors are prime!

Example 6.35

Solution

$7{x}^{3}-21{x}^{2}-70x$

Is there a GCF? Yes, $7x$.

Factor out the GCF. · $7x({x}^{2}-3x-10)$

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.

$7x(x+2)(x-5)$

$7x({x}^{2}-5x+2x-10)$

$7x({x}^{2}-3x-10)$

$7{x}^{3}-21{x}^{2}-70x\mathrm{✓}$

Be careful when you are asked to factor a binomial as there are several options!

Example 6.36

Solution

$24{y}^{2}-150$

Is there a GCF? Yes, 6.

Factor out the GCF. · $6(4{y}^{2}-25)$

In the parentheses, is it a binomial, trinomial or 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. · $6(2y-5)(2y+5)$

$\,\mathrm{Is the expression factored completely?}$

$\,\mathrm{Neither binomial can be factored.}$

Check:

$\,\mathrm{Multiply.}$

$6(2y-5)(2y+5)$

$6(4{y}^{2}-25)$

$24{y}^{2}-150\mathrm{✓}$

The next example can be factored using several methods. Recognizing the trinomial squares pattern will make your work easier.

Example 6.37

Solution

$4{a}^{2}-12ab+9{b}^{2}$

Is there a GCF? No.

Is it a binomial, trinomial, or are there more terms?

Trinomial with $a\ne{}1$. But the first term is a perfect square.

Is the last term a perfect square? Yes. · ${(2a)}^{2}-12ab+{(3b)}^{2}$ Does it fit the pattern, ${a}^{2}-2ab+{b}^{2}$? Yes. · ${(2a)}^{2}{}_{↘}\underset{-2(2a)(3b)}{-12ab+}{}_{↙}{(3b)}^{2}$ Write it as a square. · ${(2a-3b)}^{2}$

$\,\mathrm{Is the expression factored completely? Yes.}$

$\,\mathrm{The binomial cannot be factored.}$

Check your answer.

$\,\mathrm{Multiply.}$

${(2a-3b)}^{2}$

${(2a)}^{2}-2\cdot{}2a\cdot{}3b+{(3b)}^{2}$

$4{a}^{2}-12ab+9{b}^{2}\mathrm{✓}$

Remember, sums of squares do not factor, but sums of cubes do!

Example 6.38

Solution

$12{x}^{3}{y}^{2}+75x{y}^{2}$

Is there a GCF? Yes, $3x{y}^{2}$.

Factor out the GCF. · $3x{y}^{2}(4{x}^{2}+25)$

In the parentheses, is it a binomial, trinomial, or are there more than three terms? Binomial.

Is it a sum? Of squares? Yes. · Sums of squares are prime.

$\,\mathrm{Is the expression factored completely? Yes.}$

Check:

$\,\mathrm{Multiply.}$

$3x{y}^{2}(4{x}^{2}+25)$

$12{x}^{3}{y}^{2}+75x{y}^{2}\mathrm{✓}$

When using the sum or difference of cubes pattern, being careful with the signs.

Example 6.39

Solution

Example 6.40

Solution

$3{x}^{5}y-48xy$

Is there a GCF? Factor out $3xy$ · $3xy({x}^{4}-16)$ Is the binomial a sum or difference? Of squares or cubes? Write it as a difference of squares. · $3xy({({x}^{2})}^{2}-{(4)}^{2})$ Factor it as a product of conjugates · $3xy({x}^{2}-4)({x}^{2}+4)$ The first binomial is again a difference of squares. · $3xy({(x)}^{2}-{(2)}^{2})({x}^{2}+4)$ Factor it as a product of conjugates. · $3xy(x-2)(x+2)({x}^{2}+4)$

Is the expression factored completely? Yes.

Check your answer.

Multiply.

$3xy(x-2)(x+2)({x}^{2}+4)$

$3xy({x}^{2}-4)({x}^{2}+4)$

$3xy({x}^{4}-16)$

$3{x}^{5}y-48xy\mathrm{✓}$

Example 6.41

Solution

$4{x}^{2}+8bx-4ax-8ab$

Is there a GCF? Factor out the GCF, 4. · $4({x}^{2}+2bx-ax-2ab)$ There are four terms. Use grouping. · $\begin{array}{c} 4[x(x+2b)-a(x+2b)] \\ 4(x+2b)(x-a) \\ \end{array}$

Is the expression factored completely? Yes.

Check your answer.Multiply.$\,\begin{array}{c} 4(x+2b)(x-a) \\ 4({x}^{2}-ax+2bx-2ab) \\ 4{x}^{2}+8bx-4ax-8ab\mathrm{✓} \\ \end{array}$

Taking out the complete GCF in the first step will always make your work easier.

Example 6.42

Solution

$40{x}^{2}y+44xy-24y$

Is there a GCF? Factor out the GCF, $4y$. · $4y(10{x}^{2}+11x-6)$ Factor the trinomial with $a\ne{}1$. · $4y(10{x}^{2}+11x-6)$

$4y(5x-2)(2x+3)$

Is the expression factored completely? Yes.

Check your answer.

Multiply.

$4y(5x-2)(2x+3)$

$4y(10{x}^{2}+11x-6)$

$40{x}^{2}y+44xy-24y\mathrm{✓}$

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

Solution

$9{x}^{2}-12xy+4{y}^{2}-49$

Is there a GCF? No.

With more than 3 terms, use grouping. Last 2 terms have no GCF. Try grouping first 3 terms. · $9{x}^{2}-12xy+4{y}^{2}-49$

Factor the trinomial with $a\ne{}1$. But the first term is a perfect square.

Is the last term of the trinomial a perfect square? Yes. · ${(3x)}^{2}-12xy+{(2y)}^{2}-49$ Does the trinomial fit the pattern, ${a}^{2}-2ab+{b}^{2}$? Yes. · ${(3x)}^{2}{}_{↘}\underset{-2(3x)(2y)}{-12xy+}{}_{↙}{(2y)}^{2}-49$ Write the trinomial as a square. · ${(3x-2y)}^{2}-49$ Is this binomial a sum or difference? Of squares or cubes? Write it as a difference of squares. · ${(3x-2y)}^{2}-{7}^{2}$ Write 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.

$(3x-2y-7)(3x-2y+7)$

$9{x}^{2}-6xy-21x-6xy+4{y}^{2}+14y+21x-14y-49$

$9{x}^{2}-12xy+4{y}^{2}-49\mathrm{✓}$

Section 6.4 Exercises

Section 6.4 Exercises

Recognize and Use the Appropriate Method to Factor a Polynomial Completely

In the following exercises, factor completely.

Writing Exercises

ⓐ 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?

1. Factor completely: $8{y}^{3}+16{y}^{2}-24y.$

2. Factor completely: $5{y}^{3}-15{y}^{2}-270y.$

3. Factor completely: $16{x}^{3}-36x.$

1. Factor completely: $27{y}^{2}-48.$

2. Factor completely: $4{x}^{2}+20xy+25{y}^{2}.$

3. Factor completely: $9{x}^{2}-24xy+16{y}^{2}.$

1. Factor completely: $50{x}^{3}y+72xy.$

2. Factor completely: $27x{y}^{3}+48xy.$

3. Factor completely: $250{m}^{3}+432{n}^{3}.$

1. Factor completely: $2{p}^{3}+54{q}^{3}.$

2. Factor completely: $4{a}^{5}b-64ab.$

3. Factor completely: $7x{y}^{5}-7xy.$

1. Factor completely: $6{x}^{2}-12xc+6bx-12bc.$

2. Factor completely: $16{x}^{2}+24xy-4x-6y.$

3. Factor completely: $4{p}^{2}q-16pq+12q.$

1. Factor completely: $6p{q}^{2}-9pq-6p.$

2. Factor completely: $4{x}^{2}-12xy+9{y}^{2}-25.$

3. Factor completely: $16{x}^{2}-24xy+9{y}^{2}-64.$

1. $2{n}^{2}+13n-7$

2. ${a}^{5}+9{a}^{3}$

3. $121{r}^{2}-{s}^{2}$

1. $8{m}^{2}-32$

2. $25{w}^{2}-60w+36$

3. ${m}^{2}+14mn+49{n}^{2}$

1. $7{b}^{2}+7b-42$

2. $3{x}^{4}y-81xy$

3. ${k}^{4}-16$

1. $5{x}^{5}{y}^{2}-80x{y}^{2}$

2. $15pq-15p+12q-12$

3. $4{x}^{2}+40x+84$

1. $4{u}^{5}+4{u}^{2}{v}^{3}$

2. $4{c}^{2}+20cd+81{d}^{2}$

3. $10{m}^{4}-6250$

1. $36{x}^{2}y+15xy-6y$

2. $8{x}^{3}-27{y}^{3}$

3. ${y}^{6}-1$

1. $9{x}^{2}-6xy+{y}^{2}-49$

2. ${(3x+1)}^{2}-6(3x+1)+9$

3. Explain what it mean to factor a polynomial completely.

1. 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.