Elementary Algebra 2e · free preview

7.5 General Strategy for Factoring Polynomials

Learning objectives: Recognize and use the appropriate method to factor a polynomial completely

1. Learning objectives

Learning objectives

Learning objectives

  • Recognize and use the appropriate method to factor a polynomial completely

2. 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. (In your next algebra course, more methods will be added to your repertoire.) The figure below summarizes all the factoring methods we have covered. Factor polynomials. outlines a strategy you should use when factoring polynomials.

Factor 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{x}^{2}+bx+c? Undo FOIL. Is it of the form ax2+bx+ca{x}^{2}+bx+c? If aa and cc 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!

3. Example 7.59

Example 7.59

Example 7.59

Factor completely: 4x5+12x44{x}^{5}+12{x}^{4}.

Solution

IsthereaGCF?Yes, 4x4.4x5+12x4FactorouttheGCF.4x4(x+3)Intheparentheses,isitabinomial,atrinomial,oraretheremorethanthreeterms?Binomial. Isitasum?Yes. Ofsquares?Ofcubes?No.Check. Istheexpressionfactoredcompletely?Yes. Multiply. 4x4(x+3) 4x4⋅x+4x4⋅3 4x5+12x4 ✓\begin{array}{lcclccc} \mathrm{Is there a GCF?} & & & \mathrm{Yes,}\,4{x}^{4}. & & & 4{x}^{5}+12{x}^{4} \\ & & & \mathrm{Factor out the GCF.} & & & 4{x}^{4}(x+3) \\ \mathrm{In the parentheses, is it a binomial, a} & & & & & & \\ \mathrm{trinomial, or are there more than three terms?} & & & \mathrm{Binomial.} & & & \\ \,\mathrm{Is it a sum?} & & & & & & \mathrm{Yes.} \\ \,\mathrm{Of squares? Of cubes?} & & & & & & \mathrm{No.} \\ \mathrm{Check.} & & & & & & \\ \,\mathrm{Is the expression factored completely?} & & & & & & \mathrm{Yes.} \\ \,\mathrm{Multiply.} & & & & & & \\ \,4{x}^{4}(x+3) & & & & & & \\ \,4{x}^{4}\cdot{}x+4{x}^{4}\cdot{}3 & & & & & & \\ \,4{x}^{5}+12{x}^{4}\,\mathrm{✓} & & & & & & \\ \end{array}

4. Example 7.60

Example 7.60

Example 7.60

Factor completely: 12x2−11x+212{x}^{2}-11x+2.

Solution

Check.

 (3x−2)(4x−1)\,(3x-2)(4x-1)

 12x2−3x−8x+2\,12{x}^{2}-3x-8x+2

 12x2−11x+2 ✓\,12{x}^{2}-11x+2\,\mathrm{✓}

5. Example 7.61

Example 7.61

Example 7.61

Factor completely: g3+25g{g}^{3}+25g.

Solution

Is there a GCF?Yes, g.g3+25g{g}^{3}+25g
Factor out the GCF.g(g2+25)g({g}^{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.
Check.
Is the expression factored completely?Yes.
Multiply. g(g2+25) g3+25g ✓\begin{array}{l} \,g({g}^{2}+25) \\ \,{g}^{3}+25g\,\mathrm{✓} \\ \end{array}

6. Example 7.62

Example 7.62

Example 7.62

Factor completely: 12y2−7512{y}^{2}-75.

Solution

Is there a GCF?Yes, 3.12y2−7512{y}^{2}-75
Factor out the GCF.3(4y2−25)3(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.3((2y)2−(5)2)3({(2y)}^{2}-{(5)}^{2})
Write as a product of conjugates.3(2y−5)(2y+5)3(2y-5)(2y+5)
Check.
Is the expression factored completely?Yes.
Neither binomial is a difference of squares.
Multiply. 3(2y−5)(2y+5) 3(4y2−25) 12y2−75 ✓\begin{array}{l} \,3(2y-5)(2y+5) \\ \,3(4{y}^{2}-25) \\ \,12{y}^{2}-75\,\mathrm{✓} \\ \end{array}

7. Example 7.63

Example 7.63

Example 7.63

Factor completely: 4a2−12ab+9b24{a}^{2}-12ab+9{b}^{2}.

Solution

8. Example 7.64

Example 7.64

Example 7.64

Factor completely: 6y2−18y−606{y}^{2}-18y-60.

Solution

Is there a GCF?Yes, 6.6y2−18y−606{y}^{2}-18y-60
Factor out the GCF.6(y2−3y−10)6({y}^{2}-3y-10)
In the parentheses, is it a binomial, trinomial,or are there more terms?Trinomial with leading coefficient 1.
“Undo” FOIL.6(y )(y )6(y\,)(y\,)6(y+2)(y−5)6(y+2)(y-5)
Check your answer.
Is the expression factored completely?Yes.
Neither binomial is a difference of squares.
Multiply. 6(y+2)(y−5) 6(y2−5y+2y−10) 6(y2−3y−10) 6y2−18y−60 ✓\begin{array}{l} \,6(y+2)(y-5) \\ \,6({y}^{2}-5y+2y-10) \\ \,6({y}^{2}-3y-10) \\ \,6{y}^{2}-18y-60\,\mathrm{✓} \\ \end{array}

9. Example 7.65

Example 7.65

Example 7.65

Factor completely: 24x3+8124{x}^{3}+81.

Solution

10. Example 7.66

Example 7.66

Example 7.66

Factor completely: 2x4−322{x}^{4}-32.

Solution

Is there a GCF?Yes, 2.2x4−322{x}^{4}-32
Factor it out.2(x4−16)2({x}^{4}-16)
In the parentheses, is it a binomial, trinomial,or are there more than three terms?Binomial.
Is it a sum or difference?Yes.
Of squares or cubes?Difference of squares.2((x2)2−(4)2)2({({x}^{2})}^{2}-{(4)}^{2})
Write it as a product of conjugates.2(x2−4)(x2+4)2({x}^{2}-4)({x}^{2}+4)
The first binomial is again a difference of squares.2((x)2−(2)2)(x2+4)2({(x)}^{2}-{(2)}^{2})({x}^{2}+4)
Write it as a product of conjugates.2(x−2)(x+2)(x2+4)2(x-2)(x+2)({x}^{2}+4)
Is the expression factored completely?Yes.
None of these binomials is a difference of squares.
Check your answer.
Multiply. 2(x−2)(x+2)(x2+4) 2(x2−4)(x2+4) 2(x4−16) 2x4−32✓\begin{array}{l} \,2(x-2)(x+2)({x}^{2}+4) \\ \,2({x}^{2}-4)({x}^{2}+4) \\ \,2({x}^{4}-16) \\ \,2{x}^{4}-32\mathrm{✓} \\ \end{array}

11. Example 7.67

Example 7.67

Example 7.67

Factor completely: 3x2+6bx−3ax−6ab3{x}^{2}+6bx-3ax-6ab.

Solution

Is there a GCF?Yes, 3.3x2+6bx−3ax−6ab3{x}^{2}+6bx-3ax-6ab
Factor out the GCF.3(x2+2bx−ax−2ab)3({x}^{2}+2bx-ax-2ab)
In the parentheses, is it a binomial, trinomial,or are there more terms?More than 3 terms.
Use grouping.3[x(x+2b)−a(x+2b)]3(x+2b)(x−a)\begin{array}{c} 3[x(x+2b)-a(x+2b)] \\ 3(x+2b)(x-a) \\ \end{array}
Check your answer.
Is the expression factored completely? Yes.
Multiply. 3(x+2b)(x−a) 3(x2−ax+2bx−2ab) 3x2−3ax+6bx−6ab ✓\begin{array}{l} \,3(x+2b)(x-a) \\ \,3({x}^{2}-ax+2bx-2ab) \\ \,3{x}^{2}-3ax+6bx-6ab\,\mathrm{✓} \\ \end{array}

12. Example 7.68

Example 7.68

Example 7.68

Factor completely: 10x2−34x−2410{x}^{2}-34x-24.

Solution

Is there a GCF?Yes, 2.10x2−34x−2410{x}^{2}-34x-24
Factor out the GCF.2(5x2−17x−12)2(5{x}^{2}-17x-12)
In the parentheses, is it a binomial, trinomial,or are there more than three terms?Trinomial with a≠1.a\ne{}1.
Use trial and error or the “ac” method.2(5x2−17x−12)2(5x+3)(x−4)\begin{array}{c} 2(5{x}^{2}-17x-12) \\ 2(5x+3)(x-4) \\ \end{array}
Check your answer. Is the expression factoredcompletely? Yes.
Multiply. 2(5x+3)(x−4) 2(5x2−20x+3x−12) 2(5x2−17x−12) 10x2−34x−24 ✓\begin{array}{l} \,2(5x+3)(x-4) \\ \,2(5{x}^{2}-20x+3x-12) \\ \,2(5{x}^{2}-17x-12) \\ \,10{x}^{2}-34x-24\,\mathrm{✓} \\ \end{array}

13. Section 7.5 Exercises

Section 7.5 Exercises

Section 7.5 Exercises

Recognize and Use the Appropriate Method to Factor a Polynomial Completely

In the following exercises, factor completely.

Everyday Math

Writing Exercises

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ Overall, after looking at the checklist, do you think you are well-prepared for the next section? Why or why not?

14. Practice 1

  1. Factor completely: 3a4+18a33{a}^{4}+18{a}^{3}.
  1. Factor completely: 45b6+27b545{b}^{6}+27{b}^{5}.
  1. Factor completely: 10a2−17a+610{a}^{2}-17a+6.

15. Practice 2

  1. Factor completely: 8x2−18x+98{x}^{2}-18x+9.
  1. Factor completely: x3+36x{x}^{3}+36x.
  1. Factor completely: 27y2+4827{y}^{2}+48.

16. Practice 3

  1. Factor completely: 16x3−36x16{x}^{3}-36x.
  1. Factor completely: 27y2−4827{y}^{2}-48.
  1. Factor completely: 4x2+20xy+25y24{x}^{2}+20xy+25{y}^{2}.

17. Practice 4

  1. Factor completely: 9m2+42mn+49n29{m}^{2}+42mn+49{n}^{2}.
  1. Factor completely: 8y2+16y−248{y}^{2}+16y-24.
  1. Factor completely: 5u2−15u−2705{u}^{2}-15u-270.

18. Practice 5

  1. Factor completely: 250m3+432250{m}^{3}+432.
  1. Factor completely: 81q3+19281{q}^{3}+192.
  1. Factor completely: 4a4−644{a}^{4}-64.

19. Practice 6

  1. Factor completely: 7y4−77{y}^{4}-7.
  1. Factor completely: 6x2−12xc+6bx−12bc6{x}^{2}-12xc+6bx-12bc.
  1. Factor completely: 16x2+24xy−4x−6y16{x}^{2}+24xy-4x-6y.

20. Practice 7

  1. Factor completely: 4p2−16p+124{p}^{2}-16p+12.
  1. Factor completely: 6q2−9q−66{q}^{2}-9q-6.
  1. 10x4+35x310{x}^{4}+35{x}^{3}

21. Practice 8

  1. y2+10y−39{y}^{2}+10y-39
  1. 2n2+13n−72{n}^{2}+13n-7
  1. a5+9a3{a}^{5}+9{a}^{3}

22. Practice 9

  1. 121r2−s2121{r}^{2}-{s}^{2}
  1. 8m2−328{m}^{2}-32
  1. 25w2−60w+3625{w}^{2}-60w+36

23. Practice 10

  1. m2+14mn+49n2{m}^{2}+14mn+49{n}^{2}
  1. 7b2+7b−427{b}^{2}+7b-42
  1. 3x3−813{x}^{3}-81

24. Practice 11

  1. k4−16{k}^{4}-16
  1. 15pq−15p+12q−1215pq-15p+12q-12
  1. 4x2+40x+844{x}^{2}+40x+84

25. Practice 12

  1. u5+u2{u}^{5}+{u}^{2}
  1. 4c2+20cd+81d24{c}^{2}+20cd+81{d}^{2}
  1. 10m4−625010{m}^{4}-6250

26. Practice 13

  1. Watermelon drop A springtime tradition at the University of California San Diego is the Watermelon Drop, where a watermelon is dropped from the seventh story of Urey Hall.
  1. ⓐ The binomial −16t2+80-16{t}^{2}+80 gives the height of the watermelon tt seconds after it is dropped. Factor the greatest common factor from this binomial.
  1. ⓑ If the watermelon is thrown down with initial velocity 8 feet per second, its height after tt seconds is given by the trinomial −16t2−8t+80-16{t}^{2}-8t+80. Completely factor this trinomial.
  1. The difference of squares y4−625{y}^{4}-625 can be factored as (y2−25)(y2+25)({y}^{2}-25)({y}^{2}+25). But it is not completely factored. What more must be done to completely factor it?

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