# 4.1 Algebraic expressions

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## Sample set b

Identify the factors in each term.

$9{a}^{2}-6a-12$ contains three terms. Some of the factors in each term are

$\begin{array}{ll}\text{first}\text{\hspace{0.17em}}\text{term:}\hfill & 9\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}{a}^{2},\text{\hspace{0.17em}}\text{or},\text{\hspace{0.17em}}9\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}a\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}a\hfill \\ \text{second}\text{\hspace{0.17em}}\text{term:}\hfill & -6\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}a\hfill \\ \text{third}\text{\hspace{0.17em}}\text{term:}\hfill & -12\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}1,\text{\hspace{0.17em}}\text{or},\text{\hspace{0.17em}}12\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}-1\hfill \end{array}$

$14{x}^{5}y+{\left(a+3\right)}^{2}$ contains two terms. Some of the factors of these terms are

$\begin{array}{ll}\text{first}\text{\hspace{0.17em}}\text{term:}\hfill & 14,\text{\hspace{0.17em}}{x}^{5},\text{\hspace{0.17em}}y\hfill \\ \text{second}\text{\hspace{0.17em}}\text{term:}\hfill & \left(a+3\right)\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\left(a+3\right)\hfill \end{array}$

## Practice set b

In the expression $8{x}^{2}-5x+6$ , list the factors of the
first term:
second term:
third term:

8, $x$ , $x$ ; $-5$ , $x$ ; 6 and 1 or 3 and 2

In the expression $10+2\left(b+6\right){\left(b-18\right)}^{2}$ , list the factors of the
first term:
second term:

10 and 1 or 5 and 2; 2, $b+6$ , $b-18$ , $b-18$

## Common factors

Sometimes, when we observe an expression carefully, we will notice that some particular factor appears in every term. When we observe this, we say we are observing common factors . We use the phrase common factors since the particular factor we observe is common to all the terms in the expression. The factor appears in each and every term in the expression.

## Sample set c

Name the common factors in each expression.

$5{x}^{3}-7{x}^{3}+14{x}^{3}$ .

The factor ${x}^{3}$ appears in each and every term. The expression ${x}^{3}$ is a common factor.

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

The factor $x$ appears in each term. The term $4{x}^{2}$ is actually $4xx$ . Thus, $x$ is a common factor.

$12x{y}^{2}-9xy+15$ .

The only factor common to all three terms is the number 3. (Notice that $12=3\cdot 4,\text{\hspace{0.17em}}9=3\cdot 3,\text{\hspace{0.17em}}15=3\cdot 5$ .)

$3\left(x+5\right)-8\left(x+5\right)$ .

The factor $\left(x+5\right)$ appears in each term. So, $\left(x+5\right)$ is a common factor.

$45{x}^{3}{\left(x-7\right)}^{2}+15{x}^{2}\left(x-7\right)-20{x}^{2}{\left(x-7\right)}^{5}$ .

The number 5, the ${x}^{2}$ , and the $\left(x-7\right)$ appear in each term. Also, $5{x}^{2}\left(x-7\right)$ is a factor (since each of the individual quantities is joined by a multiplication sign). Thus, $5{x}^{2}\left(x-7\right)$ is a common factor.

$10{x}^{2}+9x-4$ .

There is no factor that appears in each and every term. Hence, there are no common factors in this expression.

## Practice set c

List, if any appear, the common factors in the following expressions.

${x}^{2}+5{x}^{2}-9{x}^{2}$

${x}^{2}$

$4{x}^{2}-8{x}^{3}+16{x}^{4}-24{x}^{5}$

$4{x}^{2}$

$4{\left(a+1\right)}^{3}+10\left(a+1\right)$

$2\left(a+1\right)$

$9ab\left(a-8\right)-15a{\left(a-8\right)}^{2}$

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

$14{a}^{2}{b}^{2}c\left(c-7\right)\left(2c+5\right)+28c\left(2c+5\right)$

$14c\left(2c+5\right)$

$6\left({x}^{2}-{y}^{2}\right)+19x\left({x}^{2}+{y}^{2}\right)$

no common factor

## Coefficient

In algebra, as we now know, a letter is often used to represent some quantity. Suppose we represent some quantity by the letter $x$ . The notation $5x$ means $x+x+x+x+x$ . We can now see that we have five of these quantities. In the expression $5x$ , the number 5 is called the numerical coefficient of the quantity $x$ . Often, the numerical coefficient is just called the coefficient. The coefficient of a quantity records how many of that quantity there are.

## Sample set d

$12x$ means there are $12x\text{'}\text{s}$ .

$4ab$ means there are four $ab\text{'}\text{s}$ .

$10\left(x-3\right)$ means there are ten $\left(x-3\right)\text{'}\text{s}$ .

$1y$ means there is one $y$ . We usually write just $y$ rather than $1y$ since it is clear just by looking that there is only one $y$ .

$7{a}^{3}$ means there are seven ${a}^{3\text{'}}\text{s}$ .

$5ax$ means there are five $ax\text{'}\text{s}$ . It could also mean there are $5ax\text{'}\text{s}$ . This example shows us that it is important for us to be very clear as to which quantity we are working with. When we see the expression $5ax$ we must ask ourselves "Are we working with the quantity $ax$ or the quantity $x$ ?".

$6{x}^{2}{y}^{9}$ means there are six ${x}^{2}{y}^{9\text{'}}\text{s}$ . It could also mean there are $6{x}^{2}{y}^{9\text{'}}\text{s}$ . It could even mean there are $6{y}^{9}{x}^{2\text{'}}\text{s}$ .

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