Algebraic expressions (Grade 8) — Mathematics Grades 8–11

Simplifying, expanding and substituting into algebraic expressions.

Gr 8 Terms 1 and 2 — Algebraic expressionsTerm 1 and Term 2

Practise Algebraic expressions in the app →

What gets asked

Add or subtract like terms in an expression.
Expand a bracket by multiplying every term inside it.
Substitute a number into an expression and calculate its value.
Simplify an expression using more than one operation.

You must be able to

Tell like terms apart from unlike terms.
Multiply every term inside a bracket by the term outside it.
Keep track of a negative sign when substituting a value.
Simplify an expression in the right order: brackets, then like terms.

Traps that cost marks

Treating $x$ and $x^{2}$ as like terms because they share the same letter. Like terms need the same letter and the same exponent. $x$ and $x^{2}$ cannot be combined.
Multiplying only the first term inside a bracket by the term outside it. Every term inside gets multiplied. $- x (2 x + 5) = - 2 x^{2} - 5 x$ , not $- 2 x^{2} + 5$ .
Adding the exponents when combining like terms, e.g. $3 x^{2} + 2 x^{2} = 5 x^{4}$ . Adding like terms only changes the coefficient, not the exponent. $3 x^{2} + 2 x^{2} = 5 x^{2}$ .
Losing the sign of a negative value when substituting, e.g. squaring $- 3$ as if it were $3$ . Put the negative value in brackets before you substitute: $(- 3)^{2} = 9$ .

Worked example

Expand and simplify:

3 (2 x - 1) + 4 x

Expand the bracket: $3 (2 x - 1) = 6 x - 3$ .
Add the $4 x$ : $6 x - 3 + 4 x$ .
Collect like terms: $10 x - 3$ .

Algebraic language

What the letters stand for, what the parts are called, and how to write them properly.

variable: A letter standing for a number that can change. In $18 n + 20$ the $n$ is a variable.
constant: A number in a rule that never changes. In $18 n + 20$ the constant is $20$ .
term: A piece of an expression that is added or taken away. $3 x + 5$ has two terms, $3 x$ and $5$ .
coefficient: The number in front of the letter in a term. In $7 y$ the coefficient is $7$ .
exponent: The small raised number on a letter. For a whole-number exponent it says how many times to multiply by itself.
like terms: Terms with exactly the same letters, each with the same exponent. $3 x y$ and $5 x y$ are like terms.

A letter in maths stands for a number, not for a word. A spaza shop sells bread at R18 a loaf, plus R20 to deliver. So $n$ loaves cost $18 n + 20$ rands. Here $n$ can change, so it is a variable, and $20$ never changes, so it is a constant.

Algebra has its own way of writing. Put the number in front and drop the times sign: $3 \times a$ becomes $3 a$ , never $a 3$ . Adding a letter to itself counts it twice: $x + x = 2 x$ , not $x^{2}$ .

The plus and minus signs cut an expression into terms. $2 m + 7 n - 4$ has three terms. A minus sign belongs to the term after it. But $3 x$ is one term, even though you see two symbols.

Every term has a coefficient, and may have an exponent. In $- 5 x^{2}$ the coefficient is $- 5$ and the exponent is $2$ . The sign travels with the coefficient. The exponent sits on the letter alone, so $- 5 x^{2} = - 5 \times x \times x$ .

Like terms use the same letters with the same exponents. $4 a$ and $9 a$ are like terms, and so are $3 x y$ and $5 x y$ . But $4 a$ and $9 a^{2}$ are not: the exponents differ. Nor are $4 a$ and $4 b$ .

Rules to remember

$x + x = 2 x$

Examples

A spaza shop charges

c = 18 n + 20

rands for

n

loaves. Name the variables and the constant.

Worked answer

The number of loaves can change, so $n$ is a variable.
The cost changes with it, so $c$ is a variable too.
The $20$ never moves, so it is the constant.
The $18$ is a coefficient, not a constant.

Answer: variables $c$ and $n$ ; constant $20$

A constant sits on its own. A coefficient is stuck to a letter.

Write these the way algebra wants:

b \times 7

and

k + k

Worked answer

The number goes in front and the times sign falls away: $b \times 7 = 7 b$ .
Adding $k$ to itself counts two of them: $k + k = 2 k$ .
Doubling is not squaring, so it is not $k^{2}$ .

Answer: $7 b$ and $2 k$

Both follow one habit: number in front, letter behind.

6 x^{2} - x + 9

, how many terms are there? Are any of them like terms?

Worked answer

The signs cut it into three terms: $6 x^{2}$ , $- x$ and $9$ .
The first term has coefficient $6$ and exponent $2$ .
$6 x^{2}$ and $- x$ have different exponents, so they are not like terms.

Answer: three terms; no like terms

Sharing the letter $x$ is not enough. The exponents must match too.

Traps

Calling the $18$ in $18 n + 20$ the constant. A constant sits alone with no letter. Here $18$ is a coefficient, not a constant.
Writing $x + x$ as $x^{2}$ instead of $2 x$ . Adding counts them, so $x + x = 2 x$ . Squaring would be $x \times x$ .
Giving the coefficient of $- 3 x$ as $3$ . The sign belongs to the term, so the coefficient of $- 3 x$ is $- 3$ .
Treating $x$ and $x^{2}$ as like terms because they share the same letter. Like terms need the same letter and the same exponent. $x$ and $x^{2}$ cannot be combined.

Learn and practise “Algebraic language” in the app →

Like terms and substitution

Joining the terms that can be joined, and working out what an expression is worth.

simplify: To write an expression in its shortest form, by joining every pair of terms that can be joined.
square root: The term that multiplies by itself to give it. The square root of $25 x^{2}$ is $5 x$ , since $5 x \times 5 x = 25 x^{2}$ .
substitute: To put a number in place of a letter, and then work the answer out.
numerical value: The plain number an expression comes to once every letter has a value.

Only like terms can be joined. $4 a + 9 a = 13 a$ , because you are counting how many $a$ s there are. The exponent stays exactly as it was, so $3 x^{2} + 2 x^{2} = 5 x^{2}$ .

Unlike terms stay apart. $a + b$ has to be left as it is, and so does $4 a + 9 b$ . Writing $ab$ would say something else, because $ab$ means $a \times b$ .

Every term takes its own sign with it when it moves. In $7 k - 3 m - 2 k$ the $- 2 k$ keeps its minus sign. Then $7 k - 2 k = 5 k$ , so $7 k - 3 m - 2 k = 5 k - 3 m$ .

To square a term, square the number and the letter part together: $(3 x)^{2} = 9 x^{2}$ . To cube it, use the whole term three times: $(2 x)^{3} = 8 x^{3}$ . A square root works backwards from there.

To substitute, write brackets where the letter stood, drop the number inside, then work it out. Those brackets are what keep a minus sign or an exponent attached to the right thing.

Rules to remember

$3 x^{2} + 2 x^{2} = 5 x^{2}$
Adding like terms changes the coefficient, never the exponent

Examples

Simplify:

9 p - 4 - 3 p + 7

Worked answer

$9 p$ and $- 3 p$ are like terms, so $9 p - 3 p = 6 p$ .
The plain numbers join as well: $- 4 + 7 = 3$ .
So $9 p - 4 - 3 p + 7 = 6 p + 3$ .

Answer: $6 p + 3$

Two terms were left, because letters and plain numbers cannot mix.

Simplify:

(4 y)^{2} + 2 y^{2}

Worked answer

Square both parts inside the bracket: $(4 y)^{2} = 16 y^{2}$ .
Now both terms carry the exponent $2$ , so they are like terms.
Add the coefficients: $16 y^{2} + 2 y^{2} = 18 y^{2}$ .

Answer: $18 y^{2}$

The squaring had to be finished before anything could be collected.

Find the numerical value of

- 4 x^{2}

when

x = - 3

Worked answer

Write brackets where the $x$ stood, giving $- 4 (- 3)^{2}$ .
Do the exponent first: $(- 3)^{2} = 9$ .
Then multiply: $- 4 \times 9 = - 36$ .

Answer: $- 36$

The exponent sits on $x$ alone, so the $- 4$ was never squared.

Traps

Adding the exponents when combining like terms, e.g. $3 x^{2} + 2 x^{2} = 5 x^{4}$ . Adding like terms only changes the coefficient, not the exponent. $3 x^{2} + 2 x^{2} = 5 x^{2}$ .
Turning $a + b$ into $ab$ so that it looks finished. These are unlike terms and cannot be joined. $ab = a \times b$ , which is something else.
Losing the sign of a negative value when substituting, e.g. squaring $- 3$ as if it were $3$ . Put the negative value in brackets before you substitute: $(- 3)^{2} = 9$ .
Multiplying the coefficient by the exponent, so $(- 2 x^{2})^{3}$ is given as $- 6 x^{5}$ . The coefficient is cubed as well: $(- 2 x^{2})^{3} = - 8 x^{6}$ .

Learn and practise “Like terms and substitution” in the app →

Rules in words and symbols

Reading a rule that is written in symbols, and writing one of your own from a sentence.

rule: An instruction that says what to do to a number. The rule $c = 12 n$ says multiply $n$ by $12$ .
expression: Letters and numbers written together with no equals sign, such as $3 n + 2$ .
sum: The answer when you add. The sum of $x$ and $3$ is $x + 3$ .
product: The answer when you multiply. The product of $5$ and $y$ is $5 y$ .

A letter in a rule is a number, not a label. In $c = 12 n$ the $n$ counts how many, and it is not the name of the thing being counted. Read a power just as carefully: $2^{n}$ is a power of $2$ and never $2 \times n$ .

To read a rule, put a number in and see what comes out. In $y = 4 x + 1$ , if $x$ is $3$ , then $4 \times 3 + 1 = 13$ . Do the operations in the order the rule sets out.

Going the other way, from words to symbols, is where the order trips people up. Five less than $x$ starts at $x$ and takes $5$ away, so it is $x - 5$ . The words name the $5$ first, but the $5$ is what gets removed.

When the whole of something is multiplied, you need brackets. Twice the sum of $x$ and $3$ is $2 (x + 3)$ , because the sum is doubled as one piece. Leave the brackets out and only the $x$ gets doubled.

Rules to remember

$2 (x + 3) = 2 x + 6$

Examples

The rule

c = 12 n

gives the cost in rands of

n

airtime vouchers. What does the

12

mean, and what do

7

vouchers cost?

Worked answer

The $12$ is what one voucher costs, in rands.
The $n$ is a variable that counts the vouchers.
Substitute $7$ for $n$ : $12 \times 7 = 84$ .

Answer: one voucher R $12$ ; seven vouchers R $84$

The number sitting in front of the letter is the price of one.

Write an expression for seven less than three times

k

Worked answer

Three times $k$ is the product $3 k$ .
Less than means take away, and it acts on that product.
So take $7$ off $3 k$ .

Answer: $3 k - 7$

The words name the $7$ first, but the $7$ is what gets taken away.

Write an expression for twice the sum of

m

and

5

Worked answer

The sum of $m$ and $5$ is $m + 5$ .
Twice doubles that whole sum, so brackets hold it together.
That gives $2 (m + 5)$ .
Multiplied out, $2 (m + 5) = 2 m + 10$ .

Answer: $2 (m + 5)$ , or $2 m + 10$

Without the brackets you would double only the $m$ .

Traps

Writing five less than $x$ as $5 - x$ , in the order the words come. Start at $x$ and take $5$ off it, which gives $x - 5$ .
Leaving the brackets out, so twice the sum of $x$ and $3$ becomes $2 x + 3$ . The whole sum is doubled, so it must be written $2 (x + 3)$ .
Reading $2^{n}$ as $2 \times n$ in the rule for a sequence. The $n$ is an exponent, so $2^{n}$ counts that many $2$ s multiplied together.
Treating the $m$ in a rule as a label, as though it were short for a word. A letter in a rule is a number. You can substitute a value for it.

Learn and practise “Rules in words and symbols” in the app →

Expand and simplify

Multiplying out brackets, dividing every term, and tidying up what is left.

monomial: An expression with one term, such as $3 x$ or $5 a^{2}$ .
binomial: An expression with two terms, such as $2 x + 7$ .
trinomial: An expression with three terms, such as $x^{2} + 3 x - 4$ .
expand: To multiply a bracket out so that no bracket is left. Expanding $3 (x + 2)$ gives $3 x + 6$ .

Count the terms to name an expression. One term is a monomial, like $3 x$ . Two terms make a binomial, like $2 x + 7$ . Three terms make a trinomial, like $x^{2} + 3 x - 4$ .

To multiply one monomial by another, multiply the numbers first, then the letters. When the same letter appears in both, add its exponents: $3 a \times 4 a^{2} = 12 a^{3}$ . Different letters simply stand side by side, so $2 m \times 5 n = 10 mn$ .

To expand a bracket, multiply every term inside by the term outside. Every single one, not just the first: $- x (2 x + 5) = - 2 x^{2} - 5 x$ . A trinomial works the same way: $2 (x^{2} + 3 x - 4) = 2 x^{2} + 6 x - 8$ .

Dividing runs the same way. Every term on top is divided: $\frac{6 x ^{2} + 9 x}{3} = 2 x^{2} + 3 x$ . The bottom may never be $0$ , because nothing can be shared into no parts at all.

When several operations meet, finish the multiplying and dividing first, and collect like terms only after that. So $3 (2 x - 1) + 4 x = 6 x - 3 + 4 x$ , which collects to $10 x - 3$ . A minus in front of a bracket flips every sign inside: $5 - (2 x - 3) = 8 - 2 x$ .

Rules to remember

$a (b + c) = ab + a c$
$x^{m} \times x^{n} = x^{m + n}$
Every term on top gets divided, and the bottom may never be $0$

Examples

Simplify:

2 p^{3} \times 5 p^{4}

Worked answer

Multiply the numbers first: $2 \times 5 = 10$ .
The letter is the same, so add the exponents: $3 + 4 = 7$ .
So $2 p^{3} \times 5 p^{4} = 10 p^{7}$ .

Answer: $10 p^{7}$

The exponents were added because $p$ s were being multiplied by more $p$ s.

Expand:

- 4 x (x - 3)

Worked answer

Multiply the first term inside: $- 4 x \times x = - 4 x^{2}$ .
Now the second one: $- 4 x \times (- 3) = 12 x$ .
A minus times a minus gives a plus, so that term is positive.
So $- 4 x (x - 3) = - 4 x^{2} + 12 x$ .

Answer: $- 4 x^{2} + 12 x$

Both terms inside were multiplied, and each sign was worked out on its own.

Simplify:

\frac{8 y ^{3} + 6 y ^{2}}{2 y} - 3 y

Worked answer

Divide each term on top by $2 y$ .
$\frac{8 y ^{3}}{2 y} = 4 y^{2}$ and $\frac{6 y ^{2}}{2 y} = 3 y$ .
So far it reads $4 y^{2} + 3 y - 3 y$ .
Now collect like terms: $3 y - 3 y = 0$ , leaving $4 y^{2}$ .

Answer: $4 y^{2}$

The dividing had to be finished before any terms could be collected.

Traps

Multiplying only the first term inside a bracket by the term outside it. Every term inside gets multiplied. $- x (2 x + 5) = - 2 x^{2} - 5 x$ , not $- 2 x^{2} + 5$ .
Multiplying the exponents, so $2 p^{3} \times 5 p^{4}$ is given as $10 p^{12}$ . Multiplying like bases adds the exponents: $2 p^{3} \times 5 p^{4} = 10 p^{7}$ .
Dividing only the first term on top, so $\frac{6 x ^{2} + 3}{3}$ becomes $2 x^{2} + 3$ . Every term on top is divided: $\frac{6 x ^{2} + 3}{3} = 2 x^{2} + 1$ .
Changing only the first sign when a bracket is being subtracted. The minus flips every sign inside: $5 - (2 x - 3) = 8 - 2 x$ .

Learn and practise “Expand and simplify” in the app →

Grade 8 Algebraic expressions

What gets asked

You must be able to

Traps that cost marks

Worked example

Algebraic language

Rules to remember

Examples

Traps

Like terms and substitution

Rules to remember

Examples

Traps

Rules in words and symbols

Rules to remember

Examples

Traps

Expand and simplify

Rules to remember

Examples

Traps

About this material