# 1.2 Use the language of algebra  (Page 5/18)

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Evaluate $3{x}^{2}+4x+1$ when $x=3.$

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Evaluate $6{x}^{2}-4x-7$ when $x=2.$

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## Indentify and combine like terms

Algebraic expressions are made up of terms. A term is a constant, or the product of a constant and one or more variables.

## Term

A term    is a constant, or the product of a constant and one or more variables.

Examples of terms are $7,y,5{x}^{2},9a,\text{and}\phantom{\rule{0.2em}{0ex}}{b}^{5}.$

The constant that multiplies the variable is called the coefficient .

## Coefficient

The coefficient    of a term is the constant that multiplies the variable in a term.

Think of the coefficient as the number in front of the variable. The coefficient of the term 3 x is 3. When we write x , the coefficient is 1, since $x=1·x.$

Identify the coefficient of each term: 14 y $15{x}^{2}$ a .

## Solution

The coefficient of 14 y is 14.

The coefficient of $15{x}^{2}$ is 15.

The coefficient of a is 1 since $a=1\phantom{\rule{0.2em}{0ex}}a.$

Identify the coefficient of each term: $17x$ $41{b}^{2}$ z .

14 41 1

Identify the coefficient of each term: 9 p $13{a}^{3}$ ${y}^{3}.$

9 13 1

Some terms share common traits. Look at the following 6 terms. Which ones seem to have traits in common?

$\begin{array}{cccccccccccccccc}5x\hfill & & & 7\hfill & & & {n}^{2}\hfill & & & 4\hfill & & & 3x\hfill & & & 9{n}^{2}\hfill \end{array}$

The 7 and the 4 are both constant terms.

The 5x and the 3 x are both terms with x .

The ${n}^{2}$ and the $9{n}^{2}$ are both terms with ${n}^{2}.$

When two terms are constants or have the same variable and exponent, we say they are like terms .

• 7 and 4 are like terms.
• 5 x and 3 x are like terms.
• ${x}^{2}$ and $9{x}^{2}$ are like terms.

## Like terms

Terms that are either constants or have the same variables raised to the same powers are called like terms    .

Identify the like terms: ${y}^{3},$ $7{x}^{2},$ 14, 23, $4{y}^{3},$ 9 x , $5{x}^{2}.$

## Solution

${y}^{3}$ and $4{y}^{3}$ are like terms because both have ${y}^{3};$ the variable and the exponent match.

$7{x}^{2}$ and $5{x}^{2}$ are like terms because both have ${x}^{2};$ the variable and the exponent match.

14 and 23 are like terms because both are constants.

There is no other term like 9 x .

Identify the like terms: $9,$ $2{x}^{3},$ ${y}^{2},$ $8{x}^{3},$ $15,$ $9y,$ $11{y}^{2}.$

9 and 15, ${y}^{2}$ and $11{y}^{2},$ $2{x}^{3}$ and $8{x}^{3}$

Identify the like terms: $4{x}^{3},$ $8{x}^{2},$ 19, $3{x}^{2},$ 24, $6{x}^{3}.$

19 and 24, $8{x}^{2}$ and $3{x}^{2},$ $4{x}^{3}$ and $6{x}^{3}$

Adding or subtracting terms forms an expression. In the expression $2{x}^{2}+3x+8,$ from [link] , the three terms are $2{x}^{2},3x,$ and 8.

Identify the terms in each expression.

1. $9{x}^{2}+7x+12$
2. $8x+3y$

## Solution

The terms of $9{x}^{2}+7x+12$ are $9{x}^{2},$ 7 x , and 12.

The terms of $8x+3y$ are 8 x and 3 y .

Identify the terms in the expression $4{x}^{2}+5x+17.$

$4{x}^{2},5x,17$

Identify the terms in the expression $5x+2y.$

5 x , 2 y

If there are like terms in an expression, you can simplify the expression by combining the like terms. What do you think $4x+7x+x$ would simplify to? If you thought 12 x , you would be right!

$\begin{array}{c}\hfill 4x+7x+x\hfill \\ \hfill x+x+x+x\phantom{\rule{1em}{0ex}}+x+x+x+x+x+x+x\phantom{\rule{1em}{0ex}}+x\hfill \\ \hfill 12x\hfill \end{array}$

Add the coefficients and keep the same variable. It doesn’t matter what x is—if you have 4 of something and add 7 more of the same thing and then add 1 more, the result is 12 of them. For example, 4 oranges plus 7 oranges plus 1 orange is 12 oranges. We will discuss the mathematical properties behind this later.

Simplify: $4x+7x+x.$

## How to combine like terms

Simplify: $2{x}^{2}+3x+7+{x}^{2}+4x+5.$

## Solution

Simplify: $3{x}^{2}+7x+9+7{x}^{2}+9x+8.$

$10{x}^{2}+16x+17$

Simplify: $4{y}^{2}+5y+2+8{y}^{2}+4y+5.$

$12{y}^{2}+9y+7$

## Combine like terms.

1. Identify like terms.
2. Rearrange the expression so like terms are together.
3. Add or subtract the coefficients and keep the same variable for each group of like terms.

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