# 2.EB: Exercises for Factoring Polynomial Expressions and Solving Polynomial Equations

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$

( \newcommand{\kernel}{\mathrm{null}\,}\) $$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\id}{\mathrm{id}}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\kernel}{\mathrm{null}\,}$$

$$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$

$$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$

$$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

$$\newcommand{\vectorA}[1]{\vec{#1}} % arrow$$

$$\newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$$

$$\newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vectorC}[1]{\textbf{#1}}$$

$$\newcommand{\vectorD}[1]{\overrightarrow{#1}}$$

$$\newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}$$

$$\newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}}$$

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

$$\newcommand{\avec}{\mathbf a}$$ $$\newcommand{\bvec}{\mathbf b}$$ $$\newcommand{\cvec}{\mathbf c}$$ $$\newcommand{\dvec}{\mathbf d}$$ $$\newcommand{\dtil}{\widetilde{\mathbf d}}$$ $$\newcommand{\evec}{\mathbf e}$$ $$\newcommand{\fvec}{\mathbf f}$$ $$\newcommand{\nvec}{\mathbf n}$$ $$\newcommand{\pvec}{\mathbf p}$$ $$\newcommand{\qvec}{\mathbf q}$$ $$\newcommand{\svec}{\mathbf s}$$ $$\newcommand{\tvec}{\mathbf t}$$ $$\newcommand{\uvec}{\mathbf u}$$ $$\newcommand{\vvec}{\mathbf v}$$ $$\newcommand{\wvec}{\mathbf w}$$ $$\newcommand{\xvec}{\mathbf x}$$ $$\newcommand{\yvec}{\mathbf y}$$ $$\newcommand{\zvec}{\mathbf z}$$ $$\newcommand{\rvec}{\mathbf r}$$ $$\newcommand{\mvec}{\mathbf m}$$ $$\newcommand{\zerovec}{\mathbf 0}$$ $$\newcommand{\onevec}{\mathbf 1}$$ $$\newcommand{\real}{\mathbb R}$$ $$\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}$$ $$\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}$$ $$\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}$$ $$\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}$$ $$\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$$ $$\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$$ $$\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$$ $$\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$$ $$\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}$$ $$\newcommand{\laspan}[1]{\text{Span}\{#1\}}$$ $$\newcommand{\bcal}{\cal B}$$ $$\newcommand{\ccal}{\cal C}$$ $$\newcommand{\scal}{\cal S}$$ $$\newcommand{\wcal}{\cal W}$$ $$\newcommand{\ecal}{\cal E}$$ $$\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}$$ $$\newcommand{\gray}[1]{\color{gray}{#1}}$$ $$\newcommand{\lgray}[1]{\color{lightgray}{#1}}$$ $$\newcommand{\rank}{\operatorname{rank}}$$ $$\newcommand{\row}{\text{Row}}$$ $$\newcommand{\col}{\text{Col}}$$ $$\renewcommand{\row}{\text{Row}}$$ $$\newcommand{\nul}{\text{Nul}}$$ $$\newcommand{\var}{\text{Var}}$$ $$\newcommand{\corr}{\text{corr}}$$ $$\newcommand{\len}[1]{\left|#1\right|}$$ $$\newcommand{\bbar}{\overline{\bvec}}$$ $$\newcommand{\bhat}{\widehat{\bvec}}$$ $$\newcommand{\bperp}{\bvec^\perp}$$ $$\newcommand{\xhat}{\widehat{\xvec}}$$ $$\newcommand{\vhat}{\widehat{\vvec}}$$ $$\newcommand{\uhat}{\widehat{\uvec}}$$ $$\newcommand{\what}{\widehat{\wvec}}$$ $$\newcommand{\Sighat}{\widehat{\Sigma}}$$ $$\newcommand{\lt}{<}$$ $$\newcommand{\gt}{>}$$ $$\newcommand{\amp}{&}$$ $$\definecolor{fillinmathshade}{gray}{0.9}$$

## Greatest Common Factor and Factor by Grouping

In the following exercises, find the greatest common factor.

1. $$12a^2b^3,\space 15ab^2$$

$$3ab^2$$

2. $$12m^2n^3,42m^5n^3$$

3. $$15y^3,\space 21y^2,\space 30y$$

$$3y$$

4. $$45x^3y^2,\space 15x^4y,\space 10x^5y^3$$

In the following exercises, factor the greatest common factor from each polynomial.

5. $$35y+84$$

$$7(5y+12)$$

6. $$6y^2+12y−6$$

7. $$18x^3−15x$$

$$3x(6x^2−5)$$

8. $$15m^4+6m^2n$$

9. $$4x^3−12x^2+16x$$

$$4x(x^2−3x+4)$$

10. $$−3x+24$$

11. $$−3x^3+27x^2−12x$$

$$−3x(x^2−9x+4)$$

12. $$3x(x−1)+5(x−1)$$

In the following exercises, factor by grouping.

13. $$ax−ay+bx−by$$

$$(a+b)(x−y)$$

14. $$x^2y−xy^2+2x−2y$$

15. $$x^2+7x−3x−21$$

$$(x−3)(x+7)$$

16. $$4x^2−16x+3x−12$$

17. $$m^3+m^2+m+1$$

$$(m^2+1)(m+1)$$

18. $$5x−5y−y+x$$

## Factor $$ax^2+bx+c \text{ when } a=1$$

In the following exercises, factor each trinomial completely.

1. $$a^2+14a+33$$

$$(a+3)(a+11)$$

2. $$k^2−16k+60$$

3. $$m^2+3m−54$$

$$(m+9)(m−6)$$

4. $$x^2−3x−10$$

5. $$x^2+12xy+35y^2$$

$$(x+5y)(x+7y)$$

6. $$r^2+3rs−28s^2$$

7. $$a^2+4ab−21b^2$$

$$(a+7b)(a−3b)$$

8. $$p^2−5pq−36q^2$$

9. $$m^2−5mn+30n^2$$

Prime

10. $$x^3+5x^2−24x$$

11. $$3y^3−21y^2+30y$$

$$3y(y−5)(y−2)$$

12. $$5x^4+10x^3−75x^2$$

## Factor $$ax^2+bx+c \text{ when } a\neq 1$$

In the following exercises, factor each trinomial completely.

1. $$5y^2+14y+9$$

$$(5y+9)(y+1)$$

2. $$8x^2+25x+3$$

3. $$10y^2−53y−11$$

$$(5y+1)(2y−11)$$

4. $$6p^2−19pq+10q^2$$

5. $$−81a^2+153a+18$$

$$−9(9a−1)(a+2)$$

6. $$2x^2+9x+4$$

7. $$18a^2−9a+1$$

$$(3a−1)(6a−1)$$

8. $$15p^2+2p−8$$

9. $$15x^2+6x−2$$

$$(3x−1)(5x+2)$$

10. $$8a^2+32a+24$$

11. $$3x^2+3x−36$$

$$3(x+4)(x−3)$$

12. $$48y^2+12y−36$$

13. $$18a^2−57a−21$$

$$3(2a−7)(3a+1)$$

14. $$3n^4−12n^3−96n^2$$

15. $$x^4−13x^2−30$$

$$(x^2−15)(x^2+2)$$

16. $$(x−3)^2−5(x−3)−36$$

## Factoring Special Products

In the following exercises, factor completely.

1. $$25x^2+30x+9$$

$$(5x+3)^2$$

2. $$36a^2−84ab+49b^2$$

3. $$40x^2+360x+810$$

$$10(2x+9)^2$$

4. $$5k^3−70k^2+245k$$

5. $$75u^4−30u^3v+3u^2v^2$$

$$3u^2(5u−v)^2$$

6. $$81r^2−25$$

7. $$169m^2−n^2$$

$$(13m+n)(13m−n)$$

8. $$25p^2−1$$

9. $$9−121y^2$$

$$(3+11y)(3−11y)$$

10. $$20x^2−125$$

11. $$169n^3−n$$

$$n(13n+1)(13n−1)$$

12. $$6p^2q^2−54p^2$$

13. $$24p^2+54$$

$$6(4p^2+9)$$

14. $$49x^2−81y^2$$

15. $$16z^4−1$$

$$(2z−1)(2z+1)(4z^2+1)$$

16. $$48m^4n^2−243n^2$$

17. $$a^2+6a+9−9b^2$$

$$(a+3−3b)(a+3+3b)$$

18. $$x^2−16x+64−y^2$$

19. $$a^3−125$$

$$(a−5)(a^2+5a+25)$$

20. $$b^3−216$$

21. $$2m^3+54$$

$$2(m+3)(m^2−3m+9)$$

22.$$81m^3+3$$

## General Strategy for Factoring Polynomials

In the following exercises, factor completely.

1. $$24x^3+44x^2$$

$$4x^2(6x+11)$$

2. $$24a^4−9a^3$$

3. $$16n^2−56mn+49m^2$$

$$(4n−7m)^2$$

4. $$6a^2−25a−9$$

5. $$5u^4−45u^2$$

$$5u^2(u+3)(u−3)$$

6. $$n^4−81$$

7. $$64j^2+225$$

prime

8. $$5x^2+5x−60$$

9. $$b^3−64$$

$$(b−4)(b^2+4b+16)$$

10. $$m^3+125$$

11. $$2b^2−2bc+5cb−5c^2$$

$$(2b+5c)(b−c)$$

12. $$48x^5y^2−243xy^2$$

13. $$5q^2−15q−90$$

$$5(q+3)(q−6)$$

14. $$4u^5v+4u^2v^3$$

15. $$10m^4−6250$$

$$10(m−5)(m+5)(m^2+25)$$

16. $$60x^2y−75xy+30y$$

17. $$16x^2−24xy+9y^2−64$$

$$(4x−3y+8)(4x−3y−8)$$

## Polynomial Equations

In the following exercises, solve.

1. $$(a−3)(a+7)=0$$

2. $$(5b+1)(6b+1)=0$$

$$b=−\frac{1}{5},\space b=−\frac{1}{6}$$

3. $$6m(12m−5)=0$$

4. $$(2x−1)^2=0$$

$$x=\frac{1}{2}$$

5. $$3m(2m−5)(m+6)=0$$

6. $$x^2+9x+20=0$$

$$x=−4,\space x=−5$$

7. $$y^2−y−72=0$$

8. $$2p^2−11p=40$$

$$p=−\frac{5}{2},p=8$$

9. $$q^3+3q^2+2q=0$$

10. $$144m^2−25=0$$

$$m=\frac{5}{12},\space m=−\frac{5}{12}$$

11. $$4n^2=36$$

12. $$(x+6)(x−3)=−8$$

$$x=2,\space x=−5$$

13. $$(3x−2)(x+4)=12$$

14. $$16p^3=24p^2+9p$$

$$p=0,\space p=\frac{3}{4}$$
15. $$2y^3+2y^2=12y$$