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Multiplying algebra

Simplifying expressionsFactors, multiples, powers and roots

Algebraic expressions Adding and subtracting negative numbers Multiplying and dividing negative numbersThis topic is relevant for:

Here we will learn how to expand and simplify algebraic expressions. First we expand the brackets, then we collect the like terms to simplify the expression.

At the end you’ll find expanding brackets worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

In order to expand and simplify an expression, we need to multiply out the brackets and then simplify the resulting expression by collecting the like terms.

Expanding brackets (or multiplying out) is the process by which we remove brackets.

It is the reverse process of factorisation. To expand brackets we multiply everything outside of the bracket, by everything inside of the bracket.

Once we have expanded the brackets we can simplify the expression by collecting the like terms.

E.g.

If we expand and simplify

\[2(x + 5) + 3(x − 2)\]

we will get

\[2(x + 5) + 3(x − 2) = 2x + 10 + 3x − 6\]

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‘Multiplying out brackets’ is another term for expanding brackets. It means exactly the same thing. “Expand the brackets” is the same as “multiply out the brackets”, it just gives the additional clue that when we expand brackets, we are multiplying everything outside the brackets by everything inside the brackets.

In order to expand and simplify brackets:

- Expand each bracket in the expression.
- Collect the like terms.

There are three ways to expand and simplify brackets as covered below:

- single brackets
- two or more brackets
- surds

**Expand and simplify with single brackets**.

Expand the brackets to give the following expression:

E.g. 2(x + 5) + 3(x − 1) = 2x + 10 + 3x − 3) = 5x + 7

*Remember: expressions with two terms like 5x + 7 are known as binomials.*

2**Expand and simplify with two or more brackets**.

Expand the brackets to give the following expression:

E.g. (x + 5)(x − 1) = x^{2}+ 5x − x − 5 = x^{2}+ 4x − 5

*Remember: expressions with three terms like x*

*An expression that contains more than two terms and includes variables and coefficients is called a polynomial.*

3** Expand and simplify with surds**.

E.g. (3 + √5)(2 + √5) = 6 + 3√5 + 2√5 + √5√5 = 11 + 5√5

To expand a single bracket we multiply the term outside of the bracket by everything inside of the bracket. We can simplify the expression by collecting the like terms.

Expand and simplify:

2(x + 5) + 3(x − 2)

- Expand each bracket in the expression

Multiply the first bracket:

✕ | x | + 5 |

2 | 2x | + 10 |

Multiply the second bracket – remember we are multiplying both

✕ | x | − 2 |

+ 3 | + 3x | − 6 |

*Remember to include the – sign in front of the number.*

2(x + 5) + 3(x − 2) = 2x + 10 + 3x − 6

2Collect the like terms

Highlight the two

*Remember to highlight the sign in front of the number too!*

2x + 10 + 3x − 6

2x + 3x = 5x

10 − 6 = + 4

= 5x + 4

Expand and simplify:

2x(x + 6) - 3(x - 2)

Expand each bracket in the expression.

Multiply the first bracket:

✕ | x | + 6 |

2x | 2x^{2} | + 12x |

Multiply the second bracket – remember we are multiplying both

✕ | x | − 2 |

− 3 | − 3x | + 6 |

*− ✕ − = + so − 3 ✕ − 2 gives a positive answer. We need to write + 6.*

2x(x + 6) − 3(x − 2) = 2x^{2}+ 12x − 3x + 6

Collect the like terms.

The only ‘like terms’ we have are the two

12x^{2}+ 12x - 3x + 6 12x - 3x = 9x = 12x^{2}+ 9x + 6

Expand and simplify:

3(2x − 6y) − 5(x − 2y)

Expand each bracket in the expression.

Multiply the first bracket:

✕ | 2x | − 6y |

3 | 6x | − 18y |

*+ ✕ − = − so 3 ✕ − 6y gives a negative answer. We need to write − 18y.*

Multiply the second bracket, remember we are multiplying both

✕ | x | − 2y |

− 5 | − 5x | + 10y |

*− ✕ − = + so − 5 ✕ − 2y gives a positive answer. We need to write + 10y.*

3(2x − 6y) − 5(x − 2y) = 6x − 18y − 5x + 10y

Collect the like terms.

Highlight the two

*Remember to highlight the sign in front of the number too!*

6x − 18y − 5x + 10y 6x − 5x = 1x = x − 18y + 10y = − 8y = x − 8y

1. Expand and Simplify: 3(x+7)-2(x+3)

5x+15

x+15

x+27

5x+27

3(x+7)-2(x+3)

Expand each bracket

3x+21-2x-6

Collect like terms

x+15

2. Expand and simplify: 8(y-5)-5(y-2)

13y-30

3y-50

3y-30

13y-50

8(y-5)-5(y-2)

Expand each bracket

8y-40-5y+10

Collect like terms

3y-30

3. Expand and simplify: 5x(3x-2)-4x(2x+3)

23x^{2}-22x

7x^{2}-5

7x^{2}-22x

7x^{2}-2x

5x(3x-2)-4x(2x+3)

Expand each bracket

15x^{2}-10x-8x^{2}-12x

Collect like terms

7x^{2}-22x

4. Expand and simplify: 5(6x-2y)-2(8x-5y)

14x

14x-20y

46x

14x+20y

5(6x-2y)-2(8x-5y)

Expand each bracket

30x-10y-16x+10y

Collect like terms

14x

To expand two or more brackets we multiply every term in the first bracket by every term in each of the other brackets.

Expand and simplify:

(x + 5)(x − 1)

- Expand the brackets in the expression.

✕ | x | − 1 |

x | x^{2} | − x |

+ 5 | + 5x | − 5 |

x ✕ x = x^{2}x ✕ − 1 = − x

*+ ✕ − = − so the answer is negative.*

x ✕ 5 = 5x 5 ✕ − 1 = − 5

*+ ✕ − = − so the answer is negative.*

(x + 5)(x − 1) = x^{2}− x + 5x − 5

2Collect the like terms.

The only like terms we have are the two

x^{2}− x + 5x − 5 = x^{2 }+ 4x − 5

Expand and simplify:

(2x − 4)(x + 5)

Expand the brackets in the expression.

✕ | x | + 5 |

2x | 2x^{2} | + 10x |

− 4 | − 4x | − 20 |

2x ✕ x = 2x^{2}2x ✕ 5 = 10x x ✕ − 4 = − 4x

*+ ✕ − = − so the answer is negative.*

5 ✕ − 4 = − 20

*+ ✕ − = − so the answer is negative.*

(2x − 4)(x + 5) = 2x^{2 }+ 10x − 4x − 20

Collect the like terms.

The only like terms we have are the two

2x^{2}+ 10x − 4x − 20 = 2x^{2}+ 6x − 20

Expand and simplify:

(x + 3)^{2}(x − 1)

Expand and simplify the first two brackets in the expression.

(x + 3)^{2}= (x + 3)(x + 3)

✕ | x | + 3 |

x | x^{2} | + 3x |

+ 3 | + 3x | + 9 |

x ✕ x = x^{2}x ✕ 3 = 3x x ✕ 3 = 3x 3 ✕ 3 = 9 x^{2 }+ 3x + 3x + 9 = x^{2}+ 6x + 9

Multiply this new expression with the third bracket and then simplify by collecting like terms.

✕ | x^{2} | + 6x | + 9 |

x | x^{3} | + 6x^{2} | + 9x |

− 1 | − x^{2} | − 6x | − 9 |

x ✕ x^{2 }= x^{3}x ✕ 6x = 6x^{2}x ✕ 9 = 9x − 1 ✕ x^{2 }= − x^{2}− 1 ✕ 6x = − 6x − 1 ✕ 9 = − 9 x^{3}+ 6x^{2}− x^{2}+ 9x − 6x − 9 = x^{3}+ 5x^{2}+ 3x − 9

1. Expand and simplify: (x+3)(x-7)

x^{2}-4x+21

x^{2}-10x-21

x^{2}-4x-21

x^{2}-10x+21

(x+3)(x-7)

Expand the brackets

x^{2}-7x+3x-21

and collect like terms

x^{2}-4x-21

2. Expand and simplify: (2x-4)(x-9)

2x^{2}-14x+36

2x^{2}-14x-36

2x^{2}-22x+36

2x^{2}-22x-36

(2x-4)(x-9)

Expand the brackets

2x^{2}-18x-4x+36

and collect like terms

2x^{2}-22x+36

3. Expand and simplify: (x-3)^2(2x+1)

2x^{3}-13x^{2}+12x+9

2x^{3}+13x^{2}+12x+9

2x^{3}-11x^{2}+12x+9

2x^{3}-11x^{2}-24x+9

(x-3)^{2}(2x+1)

can be written as

(x-3)(x-3)(2x+1)

Expanding the first two brackets gives

(x^{2}-3x-3x+9)(2x+1)

(x^{2}-6x+9)(2x+1)

then expanding again

2x^{3}+x^{2}-12x^{2}-6x+18x+9

and collecting like terms

2x^{3}-11x^{2}+12x+9

4. Expand and simplify: (2x-1)^{3}

8x^{3}+8x^{2}+6x+1

8x^{3}+12x^{2}+12x+1

8x^{3}-12x^{2}+6x-1

2x^{3}+12x^{2}+6x+1

(2x-1)^{3}

can be written as

(2x-1)(2x-1)(2x-1)

Expanding two of the brackets gives

(2x-1)(4x^{2}-2x-2x+1)

(2x-1)(4x^{2}-4x+1)

and expanding again

8x^{3}-8x^{2}+2x-4x^{2}+4x-1

then collect like terms

8x^{3}-12x^{2}+6x-1

To expand the brackets we need to multiply each term by every other term.

Expand and simplify:

(3 + √5)(2 + √5)

- Expand the brackets in the expression.

✕ | 2 | + √5 |

3 | 6 | + 3√5 |

+ √5 | + 2√5 | + 5 |

3 ✕ 2 = 6 3 ✕ √5 = 3√5 2 ✕ √5 = 2√5 √5 ✕ √5 = 5 = 6 + 3√5 + 2√5 + 5

2Collect the like terms.

Highlight the two constant terms

6 + 3√5 + 2√5 + 25 = 11 + 5√5

Expand and simplify:

(√2 + √5)^{2}− (3 + √5)^{2}

Expand the brackets in the expression.

Expand and simplify:

(√2 + √5)^{2}= (√2 + √5)(√2 + √5)

✕ | √2 | + √5 |

√2 | 2 | + √10 |

+ √5 | + √10 | + 5 |

2 + √10 + √10 + 5 = 7 + 2√10

Expand and simplify:

(3 + √5)^{2}= (3 + √5)(3 + √5)

✕ | 3 | + √5 |

3 | 9 | + 3√5 |

+ √5 | + 3√5 | + 5 |

9 + 3√5 + 3√5 + 5 = 14 + 6√5

Collect the like terms.

Now we will subtract the two answers.

Remember because we are taking away all of

= 7 + 2√10 − (14 + 6√5)

= 7 + 2√10 − 14 − 6√5

= − 7 + 2√10 − 6√5

1. Expand and simplify: (2+\sqrt{6})(3+\sqrt{6})\\

32+5 \sqrt{6}

12+5\sqrt{6}

12+6\sqrt{6}

7\sqrt{6}

(2+\sqrt{6})(3+\sqrt{6})\\

=6+2\sqrt{6}+3\sqrt{6}+6\\

=12+5\sqrt{6}

2. Expand and simplify: (3+\sqrt{5})(2-\sqrt{5})\\

11- \sqrt{5}

11-5\sqrt{5}

1-5\sqrt{5}

1-\sqrt{5}

(3+\sqrt{5})(2-\sqrt{5})\\

=6-3\sqrt{5}+2\sqrt{5}-5\\

=1-5\sqrt{5}

3. Expand and simplify: (\sqrt{3}+\sqrt{7})^{2}-(3+\sqrt{7})\\

13+2\sqrt{21}-\sqrt{7}

7+2\sqrt{21}+\sqrt{7}

7+\sqrt{21}-\sqrt{7}

7+2\sqrt{21}-\sqrt{7}

(\sqrt{3}+\sqrt{7})^{2}-(3+\sqrt{7})\\

=\sqrt{21}+\sqrt{21}+3+7-3-\sqrt{7}\\

=7+2\sqrt{21}-\sqrt{7}

4. Expand and simplify: (\sqrt{2}-\sqrt{8})^{2}-(\sqrt{2}+\sqrt{8})^{2}\\

16\sqrt{2}

-16

16\sqrt{8}

16

(\sqrt{2}-\sqrt{8})^{2}-(\sqrt{2}+\sqrt{8})^{2}\\

=2-2\sqrt{16}+8-2-2\sqrt{16}-8\\

=-16

**Multiplying all terms in the bracket**

We must multiply the value outside the brackets by every term inside the brackets (parentheses).

E.g.

2(6x^{2 }− 3x) = 12x^{2}− 3x ✖

Here we have not multiplied the value outside of the brackets by the second term.

The correct answer is:

2(6x^{2}− 3x) = 12x^{2}− 6x ✔

**Multiplying with negative numbers**

For two numbers to multiply to give a + their signs must be the same.

+ ✕ + = +

− ✕ − = +

e.g. 2 ✕ 3 = 6

e.g. − 2 ✕ − 3 = 6

4 ✕ 5 = 20

− 4 ✕ − 5 = 20

For two numbers to multiply to give a − their signs must be the different.

+ ✕ − = −

− ✕ + = −

e.g. 2 ✕ − 3 = − 6

e.g. − 2 ✕ 3 = − 6

4 ✕ − 5 = − 20

− 4 ✕ 5 = − 20

E.g.

− 4(3y − 5) = − 4y − 20

Here we have not used − ✕ − = +

− 4 ✕ − 5 =+ 20

So the correct answer is

**Squaring a term**

When we square something, we multiply it by itself.

E.g.

3^{2}= 3 ✕ 3 x^{2 }= x ✕ x (5y)^{2}= 5y ✕ 5y

When we square a bracket, we multiply it by the entire bracket.

(x + 3)^{2}= (x + 3)(x + 3) = x^{2}+ 6x + 9 NOT x^{2}+ 9

**Collecting like terms**

When we collect like terms we must include the sign in front of the number.

E.g.

3x + 8 + 6x − 2 NOT 3x + 8 + 6x − 2

1. Expand and simplify: – 2(y + 3)

Show answer

– 2y – 6

(1 mark)

2. Expand and simplify: 3(x – 2)+ 2(x + 5)

Show answer

5x + 4

(2 marks)

3. Expand and simplify: (2y – 3)(y + 2)

Show answer

2y^{2} + y – 6

(2 marks)

- Multiply a single term over a bracket
- Expand products of 2 or more binomials
- Simplify and manipulate algebraic expressions by collecting like terms

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