# 4A.3E: Exercises

• • OpenStax
• OpenStax
$$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$$$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

### Practice Makes Perfect

Factor Perfect Square Trinomials

In the following exercises, factor completely using the perfect square trinomials pattern.

$$16y^2+24y+9$$

$$(4y+3)^2$$

$$25v^2+20v+4$$

$$36s^2+84s+49$$

$$(6s+7)^2$$

$$49s^2+154s+121$$

$$100x^2−20x+1$$

$$(10x−1)^2$$

$$64z^2−16z+1$$

$$25n^2−120n+144$$

$$(5n−12)^2$$

$$4p^2−52p+169$$

$$49x^2+28xy+4y^2$$

$$(7x+2y)^2$$

$$25r^2+60rs+36s^2$$

$$100y^2−52y+1$$

$$(50y−1)(2y−1)$$

$$64m^2−34m+1$$

$$10jk^2+80jk+160j$$

$$10j(k+4)^2$$

$$64x^2y−96xy+36y$$

$$75u^4−30u^3v+3u^2v^2$$

$$3u^2(5u−v)^2$$

$$90p^4+300p^4q+250p^2q^2$$

Factor Differences of Squares

In the following exercises, factor completely using the difference of squares pattern, if possible.

$$25v^2−1$$

$$(5v−1)(5v+1)$$

$$169q^2−1$$

$$4−49x^2$$

$$(7x−2)(7x+2)$$

$$121−25s^2$$

$$6p^2q^2−54p^2$$

$$6p^2(q−3)(q+3)$$

$$98r^3−72r$$

$$24p^2+54$$

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

$$20b^2+140$$

$$121x^2−144y^2$$

$$(11x−12y)(11x+12y)$$

$$49x^2−81y^2$$

$$169c^2−36d^2$$

$$(13c−6d)(13c+6d)$$

$$36p^2−49q^2$$

$$16z^4−1$$

$$(2z−1)(2z+1)(4z^2+1)$$

$$m^4−n^4$$

$$162a^4b^2−32b^2$$

$$2b^2(3a−2)(3a+2)(9a^2+4)$$

$$48m^4n^2−243n^2$$

$$x^2−16x+64−y^2$$

$$(x−8−y)(x−8+y)$$

$$p^2+14p+49−q^2$$

$$a^2+6a+9−9b^2$$

$$(a+3−3b)(a+3+3b)$$

$$m^2−6m+9−16n^2$$

Factor Sums and Differences of Cubes

In the following exercises, factor completely using the sums and differences of cubes pattern, if possible.

$$x^3+125$$

$$(x+5)(x^2−5x+25)$$

$$n^6+512$$

$$z^6−27$$

$$(z^2−3)(z^4+3z^2+9)$$

$$v^3−216$$

$$8−343t^3$$

$$(2−7t)(4+14t+49t^2)$$

$$125−27w^3$$

$$8y^3−125z^3$$

$$(2y−5z)(4y^2+10yz+25z^2)$$

$$27x^3−64y^3$$

$$216a^3+125b^3$$

$$(6a+5b)(36a^2−30ab+25b^2)$$

$$27y^3+8z^3$$

$$7k^3+56$$

$$7(k+2)(k^2−2k+4)$$

$$6x^3−48y^3$$

$$2x^2−16x^2y^3$$

$$2x^2(1−2y)(1+2y+4y^2)$$

$$−2x^3y^2−16y^5$$

$$(x+3)^3+8x^3$$

$$9(x+1)(x^2+3)$$

$$(x+4)^3−27x^3$$

$$(y−5)^3−64y^3$$

$$−(3y+5)(21y^2−30y+25)$$

$$(y−5)^3+125y^3$$

Mixed Practice

In the following exercises, factor completely.

$$64a^2−25$$

$$(8a−5)(8a+5)$$

$$121x^2−144$$

$$27q^2−3$$

$$3(3q−1)(3q+1)$$

$$4p^2−100$$

$$16x^2−72x+81$$

$$(4x−9)^2$$

$$36y^2+12y+1$$

$$8p^2+2$$

$$2(4p^2+1)$$

$$81x^2+169$$

$$125−8y^3$$

$$(5−2y)(25+10y+4y^2)$$

$$27u^3+1000$$

$$45n^2+60n+20$$

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

$$48q^3−24q^2+3q$$

$$x^2−10x+25−y^2$$

$$(x+y−5)(x−y−5)$$

$$x^2+12x+36−y^2$$

$$(x+1)^3+8x^3$$

$$(3x+1)(3x^2+1)$$

$$(y−3)^3−64y^3$$

### Writing Exercises

Why was it important to practice using the binomial squares pattern in the chapter on multiplying polynomials?

How do you recognize the binomial squares pattern?

Explain why $$n^2+25\neq (n+5)^2$$. Use algebra, words, or pictures.

Maribel factored $$y^2−30y+81$$ as $$(y−9)^2$$. Was she right or wrong? How do you know?

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section. ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?

This page titled 4A.3E: Exercises is shared under a CC BY license and was authored, remixed, and/or curated by OpenStax.