# 4.1 Algebraic expressions

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$5{x}^{3}\left(y-7\right)$ means there are five ${x}^{3}\left(y-7\right)\text{'}\text{s}$ . It could also mean there are $5{x}^{3}\left(x-7\right)\text{'}\text{s}$ . It could also mean there are $5\left(x-7\right){x}^{3}\text{'}\text{s}$ .

## Practice set d

What does the coefficient of a quantity tell us?

## The difference between coefficients and exponents

It is important to keep in mind the difference between coefficients and exponents .

Coefficients record the number of like terms in an algebraic expression.
$\begin{array}{cc}\underset{4\text{\hspace{0.17em}}\text{terms}}{\underbrace{x+x+x+x}}=& \underset{\text{coefficient}\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}4}{\underset{}{4x}}\end{array}$
Exponents record the number of like factors in a term.
$\begin{array}{cc}\underset{4\text{\hspace{0.17em}}\text{factors}}{\underbrace{x\cdot x\cdot x\cdot x}}=& \underset{\text{exponent}\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}4}{\underset{}{{x}^{4}}}\end{array}$

In a term, the coefficient of a particular group of factors is the remaining group of factors.

how many of that quantity there are

## Sample set e

$3x$ .

The coefficient of $x$ is 3.

$6{a}^{3}$ .

The coefficient of ${a}^{3}$ is 6.

$9\left(4-a\right)$ .

The coefficient of $\left(4-a\right)$ is 9.

$\frac{3}{8}x{y}^{4}$ .

The coefficient of $x{y}^{4}$ is $\frac{3}{8}$ .

$3{x}^{2}y$ .

The coefficient of ${x}^{2}y$ is 3; the coefficient of $y$ is $3{x}^{2}$ ; and the coefficient of 3 is ${x}^{2}y$ .

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

The coefficient of ${\left(x+y\right)}^{2}$ is 4; the coefficient of 4 is ${\left(x+y\right)}^{2}$ ; and the coefficient of $\left(x+y\right)$ is $4\left(x+y\right)$ since $4{\left(x+y\right)}^{2}$ can be written as $4\left(x+y\right)\left(x+y\right)$ .

## Practice set e

Determine the coefficients.
In the term $6{x}^{3}$ , the coefficient of
(a) ${x}^{3}$ is .
(b) 6 is .

(a) 6 (b) ${x}^{3}$

In the term $3x\left(y-1\right)$ , the coefficient of
(a) $x\left(y-1\right)$ is .
(b) $\left(y-1\right)$ is .
(c) $3\left(y-1\right)$ is .
(d) $x$ is .
(e) 3 is .
(f) The numerical coefficient is .

(a) 3 (b) $3x$ (c) $x$ (d) $3\left(y-1\right)$ (e) $x\left(y-1\right)$ (f) 3

In the term $10a{b}^{4}$ , the coefficient of
(a) $a{b}^{4}$ is .
(b) ${b}^{4}$ is .
(c) $a$ is .
(d) 10 is .
(e) $10a{b}^{3}$ is .

(a) 10 (b) $10a$ (c) $10{b}^{4}$ (d) $a{b}^{4}$ (e) $b$

## Exercises

What is an algebraic expression?

An algebraic expression is a number, a letter, or a collection of numbers and letters along with meaningful signs of operation.

Why is the number 14 considered to be an expression?

Why is the number $x$ considered to be an expression?

$x$ is an expression because it is a letter (see the definition).

For the expressions in the following problems, write the number of terms that appear and then list the terms.

$2x+1$

$6x-10$

$\text{two}:\text{\hspace{0.17em}}\text{\hspace{0.17em}}6x,-10$

$2{x}^{3}+x-15$

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

$\text{three}:\text{\hspace{0.17em}}\text{\hspace{0.17em}}5{x}^{2},6x,-2$

$3x$

$5cz$

$\text{one}:\text{\hspace{0.17em}}\text{\hspace{0.17em}}5cz$

2

61

$\text{one}:\text{\hspace{0.17em}}\text{\hspace{0.17em}}61$

$x$

$4{y}^{3}$

$\text{one}:\text{\hspace{0.17em}}\text{\hspace{0.17em}}4{y}^{3}$

$17a{b}^{2}$

$a+1$

$\text{two}:\text{\hspace{0.17em}}\text{\hspace{0.17em}}a,1$

$\left(a+1\right)$

$2x+x+7$

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

$2x+\left(x+7\right)$

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

$\text{two}:\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(a+b\right),\left(a-1\right)$

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

For the following problems, list, if any should appear, the common factors in the expressions.

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

${x}^{2}$

$11{y}^{3}-33{y}^{3}$

$45a{b}^{2}+9{b}^{2}$

$9{b}^{2}$

$6{x}^{2}{y}^{3}+18{x}^{2}$

$2\left(a+b\right)-3\left(a+b\right)$

$\left(a+b\right)$

$8{a}^{2}\left(b+1\right)-10{a}^{2}\left(b+1\right)$

$14a{b}^{2}{c}^{2}\left(c+8\right)+12a{b}^{2}{c}^{2}$

$2a{b}^{2}{c}^{2}$

$4{x}^{2}y+5{a}^{2}b$

$9a{\left(a-3\right)}^{2}+10b\left(a-3\right)$

$\left(a-3\right)$

$15{x}^{2}-30x{y}^{2}$

$12{a}^{3}{b}^{2}c-7\left(b+1\right)\left(c-a\right)$

no commom factors

$0.06a{b}^{2}+0.03a$

$5.2{\left(a+7\right)}^{2}+17.1\left(a+7\right)$

$\left(a+7\right)$

$\frac{3}{4}{x}^{2}{y}^{2}{z}^{2}+\frac{3}{8}{x}^{2}{z}^{2}$

$\frac{9}{16}\left({a}^{2}-{b}^{2}\right)+\frac{9}{32}\left({b}^{2}-{a}^{2}\right)$

$\frac{9}{32}$

For the following problems, note how many:

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

$z\text{'}\text{s}\text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}12z?$

12

${x}^{2}\text{'s}\text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}5{x}^{2}?$

${y}^{3}\text{'s}\text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}6{y}^{3}?$

6

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

${a}^{2}b\text{'}\text{s}\text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}10{a}^{2}b?$

10

$\left(a+1\right)\text{'}\text{s}\text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}4\left(a+1\right)?$

$\left(9+y\right)\text{'}\text{s}\text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}8\left(9+y\right)?$

8

${\text{y}}^{2}\text{'s}\text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}3{x}^{3}{y}^{2}\text{?}$

$12x\text{'}\text{s}\text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}12{x}^{2}{y}^{5}?$

$x{y}^{5}$

$\left(a+5\right)\text{'}\text{s}\text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}2\left(a+5\right)?$

$\left(x-y\right)\text{'}\text{s}\text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}5x\left(x-y\right)?$

$5x$

$\left(x+1\right)\text{'}\text{s}\text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}8\left(x+1\right)?$

$2\text{'}\text{s}\text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}2{x}^{2}\left(x-7\right)?$

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

$3\left(a+8\right)\text{'}\text{s}\text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}6{x}^{2}{\left(a+8\right)}^{3}\left(a-8\right)?$

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.

$7y;\text{\hspace{0.17em}}y$

7

$10x;\text{\hspace{0.17em}}x$

$5a;\text{\hspace{0.17em}}5$

$a$

$12{a}^{2}{b}^{3}{c}^{2}{r}^{7};\text{\hspace{0.17em}}{a}^{2}{c}^{2}{r}^{7}$

$6{x}^{2}{b}^{2}\left(c-1\right);\text{\hspace{0.17em}}c-1$

$6{x}^{2}{b}^{2}$

$10x{\left(x+7\right)}^{2};\text{\hspace{0.17em}}10\left(x+7\right)$

$9{a}^{2}{b}^{5};\text{\hspace{0.17em}}3a{b}^{3}$

$3a{b}^{2}$

$15{x}^{4}{y}^{4}{\left(z+9a\right)}^{3};\text{\hspace{0.17em}}\left(z+9a\right)$

$\left(-4\right){a}^{6}{b}^{2};\text{\hspace{0.17em}}ab$

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

$\left(-11a\right){\left(a+8\right)}^{3}\left(a-1\right);\text{\hspace{0.17em}}{\left(a+8\right)}^{2}$

## Exercises for review

( [link] ) Simplify ${\left[\frac{2{x}^{8}{\left(x-1\right)}^{5}}{{x}^{4}{\left(x-1\right)}^{2}}\right]}^{4}$ .

$16{x}^{16}{\left(x-1\right)}^{12}$

( [link] ) Supply the missing phrase. Absolute value speaks to the question of and not "which way."

( [link] ) Find the value of $-\left[-6\left(-4-2\right)+7\left(-3+5\right)\right]$ .

$-50$

( [link] ) Find the value of $\frac{{2}^{5}-{4}^{2}}{{3}^{-2}}$ .

( [link] ) Express $0.0000152$ using scientific notation.

$1.52×{10}^{-5}$

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