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$5{x}^{3}(y-7)$ means there are five ${x}^{3}(y-7)\text{'}\text{s}$ . It could also mean there are $5{x}^{3}(x-7)\text{'}\text{s}$ . It could also mean there are $5(x-7){x}^{3}\text{'}\text{s}$ .
What does the coefficient of a quantity tell us?
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
$\frac{3}{8}x{y}^{4}$ .
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 ${x}^{2}y$ .
$4{(x+y)}^{2}$ .
The coefficient of ${(x+y)}^{2}$ is 4; the coefficient of 4 is ${(x+y)}^{2}$ ; and the coefficient of $(x+y)$ is $4(x+y)$ since $4{(x+y)}^{2}$ can be written as $4(x+y)(x+y)$ .
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
$3x(y-1)$ , the coefficient of
(a)
$x(y-1)$ is
(b)
$(y-1)$ is
(c)
$3(y-1)$ is
(d)
$x$ is
(e) 3 is
(f) The numerical coefficient is
(a) 3 (b) $3x$ (c) $x$ (d) $3(y-1)$ (e) $x(y-1)$ (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) $b$
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.
$6x-10$
$\text{two}:\text{\hspace{0.17em}}\text{\hspace{0.17em}}6x,-10$
$2{x}^{3}+x-15$
$5{x}^{2}+6x-2$
$\text{three}:\text{\hspace{0.17em}}\text{\hspace{0.17em}}5{x}^{2},6x,-2$
$5cz$
$\text{one}:\text{\hspace{0.17em}}\text{\hspace{0.17em}}5cz$
61
$\text{one}:\text{\hspace{0.17em}}\text{\hspace{0.17em}}61$
$4{y}^{3}$
$\text{one}:\text{\hspace{0.17em}}\text{\hspace{0.17em}}4{y}^{3}$
$17a{b}^{2}$
$a+1$
$\text{two}:\text{\hspace{0.17em}}\text{\hspace{0.17em}}a,1$
$2x+x+7$
$\text{three}:\text{\hspace{0.17em}}\text{\hspace{0.17em}}2x,x,7$
$2x+(x+7)$
$(a+1)+(a-1)$
$\text{two}:\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(a+b\right),\left(a-1\right)$
$a+1+(a-1)$
For the following problems, list, if any should appear, the common factors in the expressions.
$11{y}^{3}-33{y}^{3}$
$6{x}^{2}{y}^{3}+18{x}^{2}$
$8{a}^{2}(b+1)-10{a}^{2}(b+1)$
$14a{b}^{2}{c}^{2}(c+8)+12a{b}^{2}{c}^{2}$
$2a{b}^{2}{c}^{2}$
$4{x}^{2}y+5{a}^{2}b$
$15{x}^{2}-30x{y}^{2}$
$0.06a{b}^{2}+0.03a$
$\frac{3}{4}{x}^{2}{y}^{2}{z}^{2}+\frac{3}{8}{x}^{2}{z}^{2}$
$\frac{9}{16}({a}^{2}-{b}^{2})+\frac{9}{32}({b}^{2}-{a}^{2})$
$\frac{9}{32}$
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
$(a+1)\text{'}\text{s}\text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}4(a+1)?$
$(9+y)\text{'}\text{s}\text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}8(9+y)?$
8
${\text{y}}^{2}\text{'s}\text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}3{x}^{3}{y}^{2}\text{?}$
$12x\text{'}\text{s}\text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}12{x}^{2}{y}^{5}?$
$x{y}^{5}$
$(a+5)\text{'}\text{s}\text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}2(a+5)?$
$(x-y)\text{'}\text{s}\text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}5x(x-y)?$
$5x$
$(x+1)\text{'}\text{s}\text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}8(x+1)?$
$2\text{'}\text{s}\text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}2{x}^{2}(x-7)?$
${x}^{2}\left(x-7\right)$
$3(a+8)\text{'}\text{s}\text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}6{x}^{2}{(a+8)}^{3}(a-8)?$
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.
$10x;\text{\hspace{0.17em}}x$
$12{a}^{2}{b}^{3}{c}^{2}{r}^{7};\text{\hspace{0.17em}}{a}^{2}{c}^{2}{r}^{7}$
$6{x}^{2}{b}^{2}(c-1);\text{\hspace{0.17em}}c-1$
$6{x}^{2}{b}^{2}$
$10x{(x+7)}^{2};\text{\hspace{0.17em}}10(x+7)$
$9{a}^{2}{b}^{5};\text{\hspace{0.17em}}3a{b}^{3}$
$3a{b}^{2}$
$15{x}^{4}{y}^{4}{\left(z+9a\right)}^{3};\text{\hspace{0.17em}}\left(z+9a\right)$
$\left(-4\right){a}^{6}{b}^{2};\text{\hspace{0.17em}}ab$
$\left(-4\right){a}^{5}b$
$\left(-11a\right){\left(a+8\right)}^{3}\left(a-1\right);\text{\hspace{0.17em}}{\left(a+8\right)}^{2}$
( [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
( [link] ) Find the value of $-\left[-6(-4-2)+7(-3+5)\right]$ .
$-50$
( [link] ) Find the value of $\frac{{2}^{5}-{4}^{2}}{{3}^{-2}}$ .
( [link] ) Express $0.0000152$ using scientific notation.
$1.52\times {10}^{-5}$
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