Algebra

UK Secondary (7-11)

Components in an expression

Lesson

A mathematical expression is any calculation or formula that involves a combination of numbers and/or variables, as well as operators. Make sure you are familiar with different names for these operators so we can work mathematically with these.

Let's start with some terminology that will help us identify different parts of a mathematical expression. Some of these we will already be familiar with, and some will be new.

- Terms are parts of an expression that are separated by $+$+ or $-$− operators, or the parts of a sequence separated by commas.
- Factors are numbers or variables that can be multiplied together to get a term. For example, $2\times3=6$2×3=6, so $2$2 and $3$3 are factors of $6$6. Similarly, $5$5 and $a$
`a`are both factors of $5a$5`a`. - Coefficients are the numeric factors of terms that involve variables. We typically write the coefficient in front of the variable. For example, in the term $4x$4
`x`we would say that $4$4 is the coefficient of $x$`x`. - Constant terms are terms that don't have any variable factors. For example, in the expression $5n-4$5
`n`−4, the constant term is $-4$−4. - Like terms are terms which have exactly the same variable factors. That is to say, the variables
*and*their exponents are identical, so the only difference is the coefficients. For example, $3b$3`b`and $-17b$−17`b`are like terms. $2g^2$2`g`2 and $\frac{1}{2}g^2$12`g`2 are also like terms. Note that since constants do not have any variable factors, such as $3.5$3.5 and $-42$−42, they are also like terms!

Careful!

Algebraic terms must have the EXACT SAME combination of variables to be like terms.

- $9ef^2$9
`e``f`2 and $10ef^2$10`e``f`2 are like terms because they both have variables of $e$`e`and $f^2$`f`2. - Similarly, $4st$4
`s``t`and $-5ts$−5`t``s`are like terms because they both have variables of $s$`s`and $t$`t`, even though they are written in a different order. - $xy$
`x``y`and $xy^2$`x``y`2 ARE NOT like terms, however, because the powers of $y$`y`are different between the two terms.

This picture summarises some of this terminology:

There are a few conventions that we make when writing algebraic expressions, some of which we have already been using.

- When we multiply two numbers together, we use a multiplication sign, such as $2\times3$2×3. When we multiply a number by a variable, or when we multiply variables together, we leave out the multiplication sign. So $2\times y$2×
`y`is written as $2y$2`y`, for example, and $a^2\times b$`a`2×`b`is written as $a^2b$`a`2`b`. - When we multiply a number by one or more variables, we write the number first and then the variables. For example, $p\times3\times q$
`p`×3×`q`would be written as $3pq$3`p``q`. - If we multiply one or more variables by $1$1, we can leave off the $1$1. For example, instead of writing $1x$1
`x`or $1\times x$1×`x`, we can just write $x$`x`. - We usually avoid using the division symbol \div, and instead write division using fractions. So instead of writing $12\div t$12÷
`t`, we would write $\frac{12}{t}$12`t`. This helps to avoid confusion about the order of operations in an expression. - If we multiply a variable by itself, we usually simplify the expression by using an exponent. So if we have the expression $m\times m\times m$
`m`×`m`×`m`, we would write $m^3$`m`3 instead of $mmm$`m``m``m`.

The expression $2+1+8$2+1+8 represents the number of rockmelons, apples, and lemons in Caitlin's shopping trolley

a) What type of expression does $2+1+8$2+1+8 represent?

A) Sum B) Product C) Quotient D) Difference

Think: Which of these words means that the terms are *added*?

Do: This expression represents a sum.

b) How many terms does the expression have?

Think: Terms are separated by $+$+ and $-$− signs. How many numbers are being separated by the plus signs here?

Do: There are $3$3 terms in this expression.

The expression $7y$7`y` represents the total number of biscuits in $y$`y` packets if each packet contains $7$7 biscuits.

a) What type of expression does $7y$7`y` represent?

A) Product B) Quotient C) Difference D) Sum

Think: Which of these words means that the terms are *multiplied*?

Do: This expression is a product.

b) What are the factors in the expression?

Think: Factors are numbers or variables that are multiplied together to get a term.

Do: $7y$7`y` means $7\times y$7×`y`. So we can see that the factors in this expression are $7$7 and $y$`y`.

Rewrite the expression $z\div5$`z`÷5 without using a division sign.

Think: What conventions do we use in algebraic expressions?

Do: We can write expressions that involve division by using fractions. So $z\div5$`z`÷5 is the same as $\frac{z}{5}$`z`5.

Consider the expression $-5+3a-6+8a$−5+3`a`−6+8`a`.

a) How many terms are in the expression?

Think: Terms are separated by $+$+ and $-$− signs. Let's count them.

Do: There are $4$4 terms in this expression. (They are $-5$−5, $3a$3`a`, $-6$−6 and $8a$8`a`).

b) What is the first term?

Think: What term is written first in this expression?

Do: The first term is $-5$−5.

c) What are the like terms?

Think: Like terms are terms with exactly same variable parts.

Do: There are two pairs of like terms in this expression. The first pair of like terms is $-5$−5 and $-6$−6. The second pair of like terms is $3a$3`a` and $8a$8`a`.

d) What are the coefficients?

Think: Coefficients are numbers that come before variable factors.

Do: There are two coefficients in this expression: $3$3 and $8$8.

e) What are the constant terms?

Think: Constant terms are terms without any variables.

Do: There are two constants in this expression: $-5$−5 and $-6$−6.

The expression $n+9$`n`+9 represents nine more than the number of points, $n$`n`, scored by the opposition team in a basketball game.

a) What type of expression does $n+9$`n`+9 represent?

b) How many terms does the expression have?

c) What is the constant term?

The expression $8u+2$8`u`+2 represents the cost of a $u$`u`-minute international phone call, where $2$2 represents the connection cost.

a) How many terms are there in the expression?

b) What is the second term?

c) Identify the coefficient.

d) What is the constant term?