# 4.9 Exercise supplement

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This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. Operations with algebraic expressions and numerical evaluations are introduced in this chapter. Coefficients are described rather than merely defined. Special binomial products have both literal and symbolic explanations and since they occur so frequently in mathematics, we have been careful to help the student remember them. In each example problem, the student is "talked" through the symbolic form.This module provides an exercise supplement for the chapter "Algebraic Expressions and Equations".

## Algebraic expressions ( [link] )

For the following problems, write the number of terms that appear, then write the terms.

$4{x}^{2}+7x+12$

$\text{three:\hspace{0.17em}}4{x}^{2},7x,12$

$14{y}^{6}$

$c+8$

$\text{two:\hspace{0.17em}}c,8$

$8$

List, if any should appear, the common factors for the following problems.

${a}^{2}+4{a}^{2}+6{a}^{2}$

${a}^{2}$

$9{y}^{4}-18{y}^{4}$

$12{x}^{2}{y}^{3}+36{y}^{3}$

$12{y}^{3}$

$6\left(a+4\right)+12\left(a+4\right)$

$4\left(a+2b\right)+6\left(a+2b\right)$

$2\left(a+2b\right)$

$17{x}^{2}y\left(z+4\right)+51y\left(z+4\right)$

$6{a}^{2}{b}^{3}c+5{x}^{2}y$

no common factors

For the following problems, answer the question of how many.

$x\text{'}\text{s}\text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}9x?$

$\left(a+b\right)\text{'}\text{s}\text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}\text{12}\left(a+b\right)?$

12

${a}^{4}\text{'}\text{s}\text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}6{a}^{4}?$

${c}^{3}\text{'}\text{s}\text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}2{a}^{2}b{c}^{3}?$

$2{a}^{2}b$

${\left(2x+3y\right)}^{2}\text{'}\text{s}\text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}5\left(x+2y\right){\left(2x+3y\right)}^{3}?$

For the following problems, a term will be given followed by a group of its factors. List the coefficient of the given group of factors.

$8z,\text{\hspace{0.17em}}z$

8

$16{a}^{3}{b}^{2}{c}^{4},\text{\hspace{0.17em}}{c}^{4}$

$7y\left(y+3\right),\text{\hspace{0.17em}}7y$

$\left(y+3\right)$

$\left(-5\right){a}^{5}{b}^{5}{c}^{5},\text{\hspace{0.17em}}bc$

For the following problems, observe the equations and write the relationship being expressed.

$a=3b$

$\text{The\hspace{0.17em}value\hspace{0.17em}of\hspace{0.17em}}a\text{\hspace{0.17em}is\hspace{0.17em}equal\hspace{0.17em}to\hspace{0.17em}three\hspace{0.17em}time\hspace{0.17em}the\hspace{0.17em}value\hspace{0.17em}of\hspace{0.17em}}b\text{.}$

$r=4t+11$

$f=\frac{1}{2}{m}^{2}+6g$

$\text{The\hspace{0.17em}value\hspace{0.17em}of\hspace{0.17em}}f\text{\hspace{0.17em}is\hspace{0.17em}equal\hspace{0.17em}to\hspace{0.17em}six\hspace{0.17em}times\hspace{0.17em}}g\text{\hspace{0.17em}more\hspace{0.17em}then\hspace{0.17em}one\hspace{0.17em}half\hspace{0.17em}time\hspace{0.17em}the\hspace{0.17em}value\hspace{0.17em}of\hspace{0.17em}}m\text{\hspace{0.17em}squared}\text{.}$

$x=5{y}^{3}+2y+6$

${P}^{2}=k{a}^{3}$

$\text{The\hspace{0.17em}value\hspace{0.17em}of\hspace{0.17em}}P\text{\hspace{0.17em}squared\hspace{0.17em}is\hspace{0.17em}equal\hspace{0.17em}to\hspace{0.17em}the\hspace{0.17em}value\hspace{0.17em}of\hspace{0.17em}}a\text{\hspace{0.17em}cubed\hspace{0.17em}times\hspace{0.17em}}k\text{.}$

Use numerical evaluation to evaluate the equations for the following problems.

$\begin{array}{ll}C=2\pi r.\hfill & \text{Find}\text{\hspace{0.17em}}C\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}\pi \text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{approximated}\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}\hfill \\ \hfill & 3.14\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}r=6.\hfill \end{array}$

$\begin{array}{cc}I=\frac{E}{R}.& \text{Find}\text{\hspace{0.17em}}I\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}E=20\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}R=2.\end{array}$

10

$\begin{array}{cc}I=prt.& \text{Find}\text{\hspace{0.17em}}I\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}p=1000,\text{\hspace{0.17em}}r=0.06,\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}t=3.\end{array}$

$\begin{array}{cc}E=m{c}^{2}.& \text{Find}\text{\hspace{0.17em}}E\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}m=120\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}c=186,000.\end{array}$

$\text{4}\text{.1515}×{\text{10}}^{12}$

$\begin{array}{ll}z=\frac{x-u}{s}.\hfill & \text{Find}\text{\hspace{0.17em}}z\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}x=42,\text{\hspace{0.17em}}u=30,\text{\hspace{0.17em}}\text{and}\hfill \\ s=12.\hfill & \hfill \end{array}$

$\begin{array}{ll}R=\frac{24C}{P\left(n+1\right)}.\hfill & \text{Find}\text{\hspace{0.17em}}R\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}C=35,\text{\hspace{0.17em}}P=300,\text{\hspace{0.17em}}\text{and}\hfill \\ n=19.\hfill & \hfill \end{array}$

$\frac{7}{50}\text{\hspace{0.17em}}\text{or}\text{\hspace{0.17em}}0.14$

## Classification of expressions and equations ( [link] )

For the following problems, classify each of the polynomials as a monomial, binomial, or trinomial. State the degree of each polynomial and write the numerical coefficient of each term.

$2a+9$

$4{y}^{3}+3y+1$

trinomial; cubic; 4, 3, 1

$10{a}^{4}$

$147$

monomial; zero; 147

$4xy+2y{z}^{2}+6x$

$9a{b}^{2}{c}^{2}+10{a}^{3}{b}^{2}{c}^{5}$

binomial; tenth; 9, 10

$\begin{array}{cc}{\left(2x{y}^{3}\right)}^{0},& x{y}^{3}\ne 0\end{array}$

Why is the expression $\frac{4x}{3x-7}$ not a polynomial?

. . . because there is a variable in the denominator

Why is the expression $5{a}^{3/4}$ not a polynomial?

For the following problems, classify each of the equations by degree. If the term linear, quadratic, or cubic applies, use it.

$3y+2x=1$

linear

$4{a}^{2}-5a+8=0$

$y-x-z+4w=21$

linear

$5{x}^{2}+2{x}^{2}-3x+1=19$

${\left(6{x}^{3}\right)}^{0}+5{x}^{2}=7$

## Combining polynomials using addition and subtraction ( [link] ) - special binomial products ( [link] )

Simplify the algebraic expressions for the following problems.

$4{a}^{2}b+8{a}^{2}b-{a}^{2}b$

$21{x}^{2}{y}^{3}+3xy+{x}^{2}{y}^{3}+6$

$22{x}^{2}{y}^{3}+3xy+6$

$7\left(x+1\right)+2x-6$

$2\left(3{y}^{2}+4y+4\right)+5{y}^{2}+3\left(10y+2\right)$

$11{y}^{2}+38y+14$

$5\left[3x+7\left(2{x}^{2}+3x+2\right)+5\right]-10{x}^{2}+4\left(3{x}^{2}+x\right)$

$8\left\{3\left[4{y}^{3}+y+2\right]+6\left({y}^{3}+2{y}^{2}\right)\right\}-24{y}^{3}-10{y}^{2}-3$

$120{y}^{3}+86{y}^{2}+24y+45$

$4{a}^{2}b{c}^{3}+5ab{c}^{3}+9ab{c}^{3}+7{a}^{2}b{c}^{2}$

$x\left(2x+5\right)+3{x}^{2}-3x+3$

$5{x}^{2}+2x+3$

$4k\left(3{k}^{2}+2k+6\right)+k\left(5{k}^{2}+k\right)+16$

$2\left\{5\left[6\left(b+2a+{c}^{2}\right)\right]\right\}$

$60{c}^{2}+120a+60b$

$9{x}^{2}y\left(3xy+4x\right)-7{x}^{3}{y}^{2}-30{x}^{3}y+5y\left({x}^{3}y+2x\right)$

$3m\left[5+2m\left(m+6{m}^{2}\right)\right]+m\left({m}^{2}+4m+1\right)$

$36{m}^{4}+7{m}^{3}+4{m}^{2}+16m$

$2r\left[4\left(r+5\right)-2r-10\right]+6r\left(r+2\right)$

$abc\left(3abc+c+b\right)+6a\left(2bc+b{c}^{2}\right)$

$3{a}^{2}{b}^{2}{c}^{2}+7ab{c}^{2}+a{b}^{2}c+12abc$

${s}^{10}\left(2{s}^{5}+3{s}^{4}+4{s}^{3}+5{s}^{2}+2s+2\right)-{s}^{15}+2{s}^{14}+3s\left({s}^{12}+4{s}^{11}\right)-{s}^{10}$

$6{a}^{4}\left({a}^{2}+5\right)$

$6{a}^{6}+30{a}^{4}$

$2{x}^{2}{y}^{4}\left(3{x}^{2}y+4xy+3y\right)$

$5{m}^{6}\left(2{m}^{7}+3{m}^{4}+{m}^{2}+m+1\right)$

$10{m}^{13}+15{m}^{10}+5{m}^{8}+5{m}^{7}+5{m}^{6}$

${a}^{3}{b}^{3}{c}^{4}\left(4a+2b+3c+ab+ac+b{c}^{2}\right)$

$\left(x+2\right)\left(x+3\right)$

${x}^{2}+5x+6$

$\left(y+4\right)\left(y+5\right)$

$\left(a+1\right)\left(a+3\right)$

${a}^{2}+4a+3$

$\left(3x+4\right)\left(2x+6\right)$

$4xy-10xy$

$-6xy$

$5a{b}^{2}-3\left(2a{b}^{2}+4\right)$

$7{x}^{4}-15{x}^{4}$

$-8{x}^{4}$

$5{x}^{2}+2x-3-7{x}^{2}-3x-4-2{x}^{2}-11$

$4\left(x-8\right)$

$4x-32$

$7x\left({x}^{2}-x+3\right)$

$-3a\left(5a-6\right)$

$-15{a}^{2}+18a$

$4{x}^{2}{y}^{2}\left(2x-3y-5\right)-16{x}^{3}{y}^{2}-3{x}^{2}{y}^{3}$

$-5y\left({y}^{2}-3y-6\right)-2y\left(3{y}^{2}+7\right)+\left(-2\right)\left(-5\right)$

$-11{y}^{3}+15{y}^{2}+16y+10$

$-\left[-\left(-4\right)\right]$

$-\left[-\left(-\left\{-\left[-\left(5\right)\right]\right\}\right)\right]$

$-5$

${x}^{2}+3x-4-4{x}^{2}-5x-9+2{x}^{2}-6$

$4{a}^{2}b-3{b}^{2}-5{b}^{2}-8{q}^{2}b-10{a}^{2}b-{b}^{2}$

$-6{a}^{2}b-8{q}^{2}b-9{b}^{2}$

$2{x}^{2}-x-\left(3{x}^{2}-4x-5\right)$

$3\left(a-1\right)-4\left(a+6\right)$

$-a-27$

$-6\left(a+2\right)-7\left(a-4\right)+6\left(a-1\right)$

Add $-3x+4$ to $5x-8$ .

$2x-4$

Add $4\left({x}^{2}-2x-3\right)$ to $-6\left({x}^{2}-5\right)$ .

Subtract 3 times $\left(2x-1\right)$ from 8 times $\left(x-4\right)$ .

$2x-29$

$\left(x+4\right)\left(x-6\right)$

$\left(x-3\right)\left(x-8\right)$

${x}^{2}-11x+24$

$\left(2a-5\right)\left(5a-1\right)$

$\left(8b+2c\right)\left(2b-c\right)$

$16{b}^{2}-4bc-2{c}^{2}$

${\left(a-3\right)}^{2}$

${\left(3-a\right)}^{2}$

${a}^{2}-6a+9$

${\left(x-y\right)}^{2}$

${\left(6x-4\right)}^{2}$

$36{x}^{2}-48x+16$

${\left(3a-5b\right)}^{2}$

${\left(-x-y\right)}^{2}$

${x}^{2}+2xy+{y}^{2}$

$\left(k+6\right)\left(k-6\right)$

$\left(m+1\right)\left(m-1\right)$

${m}^{2}-1$

$\left(a-2\right)\left(a+2\right)$

$\left(3c+10\right)\left(3c-10\right)$

$9{c}^{2}-100$

$\left(4a+3b\right)\left(4a-3b\right)$

$\left(5+2b\right)\left(5-2b\right)$

$25-4{b}^{2}$

$\left(2y+5\right)\left(4y+5\right)$

$\left(y+3a\right)\left(2y+a\right)$

$2{y}^{2}+7ay+3{a}^{2}$

$\left(6+a\right)\left(6-3a\right)$

$\left({x}^{2}+2\right)\left({x}^{2}-3\right)$

${x}^{4}-{x}^{2}-6$

$6\left(a-3\right)\left(a+8\right)$

$8\left(2y-4\right)\left(3y+8\right)$

$48{y}^{2}+32y-256$

$x\left(x-7\right)\left(x+4\right)$

${m}^{2}n\left(m+n\right)\left(m+2n\right)$

${m}^{4}n+3{m}^{3}{n}^{2}+2{m}^{2}{n}^{3}$

$\left(b+2\right)\left({b}^{2}-2b+3\right)$

$3p\left({p}^{2}+5p+4\right)\left({p}^{2}+2p+7\right)$

$3{p}^{5}+21{p}^{4}+63{p}^{3}+129{p}^{2}+84p$

${\left(a+6\right)}^{2}$

${\left(x-2\right)}^{2}$

${x}^{2}-4x+4$

${\left(2x-3\right)}^{2}$

${\left({x}^{2}+y\right)}^{2}$

${x}^{4}+2{x}^{2}y+{y}^{2}$

${\left(2m-5n\right)}^{2}$

${\left(3{x}^{2}{y}^{3}-4{x}^{4}y\right)}^{2}$

$9{x}^{4}{y}^{6}-24{x}^{6}{y}^{4}+16{x}^{8}{y}^{2}$

${\left(a-2\right)}^{4}$

## Terminology associated with equations ( [link] )

Find the domain of the equations for the following problems.

$y=8x+7$

all real numbers

$y=5{x}^{2}-2x+6$

$y=\frac{4}{x-2}$

all real numbers except 2

$m=\frac{-2x}{h}$

$z=\frac{4x+5}{y+10}$

$x\text{\hspace{0.17em}can\hspace{0.17em}equal\hspace{0.17em}any\hspace{0.17em}real\hspace{0.17em}number;\hspace{0.17em}}y\text{\hspace{0.17em}can\hspace{0.17em}equal\hspace{0.17em}any\hspace{0.17em}number\hspace{0.17em}except}-10$

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