Intermediate Algebra

# 6.4General Strategy for Factoring Polynomials

Intermediate Algebra6.4 General Strategy for Factoring Polynomials

### Learning Objectives

By the end of this section, you will be able to:
• 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.

### How To

#### Use a general strategy for factoring polynomials.

1. Step 1. Is there a greatest common factor?
Factor it out.
2. 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.
If it is a trinomial:
• Is it of the form $x2+bx+c?x2+bx+c?$ Undo FOIL.
• Is it of the form $ax2+bx+c?ax2+bx+c?$
If a and c are squares, check if it fits the trinomial square pattern.
Use the trial and error or “ac” method.
If it has more than three terms:
• Use the grouping method.
3. 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: $7x3−21x2−70x.7x3−21x2−70x.$

### Try It 6.69

Factor completely: $8y3+16y2−24y.8y3+16y2−24y.$

### Try It 6.70

Factor completely: $5y3−15y2−270y.5y3−15y2−270y.$

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

### Example 6.36

Factor completely: $24y2−150.24y2−150.$

### Try It 6.71

Factor completely: $16x3−36x.16x3−36x.$

### Try It 6.72

Factor completely: $27y2−48.27y2−48.$

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

### Example 6.37

Factor completely: $4a2−12ab+9b2.4a2−12ab+9b2.$

### Try It 6.73

Factor completely: $4x2+20xy+25y2.4x2+20xy+25y2.$

### Try It 6.74

Factor completely: $9x2−24xy+16y2.9x2−24xy+16y2.$

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

### Example 6.38

Factor completely $12x3y2+75xy2.12x3y2+75xy2.$

### Try It 6.75

Factor completely: $50x3y+72xy.50x3y+72xy.$

### Try It 6.76

Factor completely: $27xy3+48xy.27xy3+48xy.$

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

### Example 6.39

Factor completely: $24x3+81y3.24x3+81y3.$

### Try It 6.77

Factor completely: $250m3+432n3.250m3+432n3.$

### Try It 6.78

Factor completely: $2p3+54q3.2p3+54q3.$

### Example 6.40

Factor completely: $3x5y−48xy.3x5y−48xy.$

### Try It 6.79

Factor completely: $4a5b−64ab.4a5b−64ab.$

### Try It 6.80

Factor completely: $7xy5−7xy.7xy5−7xy.$

### Example 6.41

Factor completely: $4x2+8bx−4ax−8ab.4x2+8bx−4ax−8ab.$

### Try It 6.81

Factor completely: $6x2−12xc+6bx−12bc.6x2−12xc+6bx−12bc.$

### Try It 6.82

Factor completely: $16x2+24xy−4x−6y.16x2+24xy−4x−6y.$

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

### Example 6.42

Factor completely: $40x2y+44xy−24y.40x2y+44xy−24y.$

### Try It 6.83

Factor completely: $4p2q−16pq+12q.4p2q−16pq+12q.$

### Try It 6.84

Factor completely: $6pq2−9pq−6p.6pq2−9pq−6p.$

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: $9x2−12xy+4y2−49.9x2−12xy+4y2−49.$

### Try It 6.85

Factor completely: $4x2−12xy+9y2−25.4x2−12xy+9y2−25.$

### Try It 6.86

Factor completely: $16x2−24xy+9y2−64.16x2−24xy+9y2−64.$

### Section 6.4 Exercises

#### Practice Makes Perfect

Recognize and Use the Appropriate Method to Factor a Polynomial Completely

In the following exercises, factor completely.

233.

$2 n 2 + 13 n − 7 2 n 2 + 13 n − 7$

234.

$8 x 2 − 9 x − 3 8 x 2 − 9 x − 3$

235.

$a 5 + 9 a 3 a 5 + 9 a 3$

236.

$75 m 3 + 12 m 75 m 3 + 12 m$

237.

$121 r 2 − s 2 121 r 2 − s 2$

238.

$49 b 2 − 36 a 2 49 b 2 − 36 a 2$

239.

$8 m 2 − 32 8 m 2 − 32$

240.

$36 q 2 − 100 36 q 2 − 100$

241.

$25 w 2 − 60 w + 36 25 w 2 − 60 w + 36$

242.

$49 b 2 − 112 b + 64 49 b 2 − 112 b + 64$

243.

$m 2 + 14 m n + 49 n 2 m 2 + 14 m n + 49 n 2$

244.

$64 x 2 + 16 x y + y 2 64 x 2 + 16 x y + y 2$

245.

$7 b 2 + 7 b − 42 7 b 2 + 7 b − 42$

246.

$30 n 2 + 30 n + 72 30 n 2 + 30 n + 72$

247.

$3 x 4 y − 81 x y 3 x 4 y − 81 x y$

248.

$4 x 5 y − 32 x 2 y 4 x 5 y − 32 x 2 y$

249.

$k 4 − 16 k 4 − 16$

250.

$m 4 − 81 m 4 − 81$

251.

$5 x 5 y 2 − 80 x y 2 5 x 5 y 2 − 80 x y 2$

252.

$48 x 5 y 2 − 243 x y 2 48 x 5 y 2 − 243 x y 2$

253.

$15 p q − 15 p + 12 q − 12 15 p q − 15 p + 12 q − 12$

254.

$12 a b − 6 a + 10 b − 5 12 a b − 6 a + 10 b − 5$

255.

$4 x 2 + 40 x + 84 4 x 2 + 40 x + 84$

256.

$5 q 2 − 15 q − 90 5 q 2 − 15 q − 90$

257.

$4 u 5 + 4 u 2 v 3 4 u 5 + 4 u 2 v 3$

258.

$5 m 4 n + 320 m n 4 5 m 4 n + 320 m n 4$

259.

$4 c 2 + 20 c d + 81 d 2 4 c 2 + 20 c d + 81 d 2$

260.

$25 x 2 + 35 x y + 49 y 2 25 x 2 + 35 x y + 49 y 2$

261.

$10 m 4 − 6250 10 m 4 − 6250$

262.

$3 v 4 − 768 3 v 4 − 768$

263.

$36 x 2 y + 15 x y − 6 y 36 x 2 y + 15 x y − 6 y$

264.

$60 x 2 y − 75 x y + 30 y 60 x 2 y − 75 x y + 30 y$

265.

$8 x 3 − 27 y 3 8 x 3 − 27 y 3$

266.

$64 x 3 + 125 y 3 64 x 3 + 125 y 3$

267.

$y 6 − 1 y 6 − 1$

268.

$y 6 + 1 y 6 + 1$

269.

$9 x 2 − 6 x y + y 2 − 49 9 x 2 − 6 x y + y 2 − 49$

270.

$16 x 2 − 24 x y + 9 y 2 − 64 16 x 2 − 24 x y + 9 y 2 − 64$

271.

$( 3 x + 1 ) 2 − 6 ( 3 x + 1 ) + 9 ( 3 x + 1 ) 2 − 6 ( 3 x + 1 ) + 9$

272.

$( 4 x − 5 ) 2 − 7 ( 4 x − 5 ) + 12 ( 4 x − 5 ) 2 − 7 ( 4 x − 5 ) + 12$

#### Writing Exercises

273.

Explain what it mean to factor a polynomial completely.

274.

The difference of squares $y4−625y4−625$ can be factored as $(y2−25)(y2+25).(y2−25)(y2+25).$ But it is not completely factored. What more must be done to completely factor.

275.

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.

276.

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?

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