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Expanding brackets and factorising


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Algebra

Expanding Brackets and Factorising

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This topic will cover how to:

  • expand either one or two sets of brackets in an algebraic expression;
  • factorise an algebraic expression using one set of brackets; and
  • factorise a particular type of quadratic using two sets of brackets.

Expanding One Set of Brackets

Sometimes we have an algebraic expression that contains brackets, and we may wish to remove the brackets to work with the expression.

One example of this is when we have a term in front of a set of brackets, and an algebraic expression inside them. To expand the brackets in this case we multiply each term inside the brackets by the term outside, which can be done by following the steps below:

1.Write out each of the multiplications you need to perform, by putting the term outside the brackets together with each term inside the brackets and adding these terms together.
For example, when expanding
-3a(2b + 4a - 9) write out (-3a)(2b) + (-3a)(4a) + (-3a)(-9)

2.Perform each of the required multiplications, leaving each result in brackets in order to keep track of any negative signs. These will be dealt with in the next step.
For example, multiplying pairs of terms in
(-3a)(2b) + (-3a)(4a) + (-3a)(-9) gives (-6ab) + (-12a2) + (27a)

3.Remove the brackets from each term, replacing a positive sign with a negative sign if there is one inside the corresponding set of brackets.
For example, removing brackets and using appropriate signs for
(-6ab) + (-12a2) + (27a) gives -6ab - 12a2 + 27a

Expanding Pairs of Brackets

Another example of expanding brackets is when we have two lots of brackets that are being multiplied together.

To expand the brackets in this case we multiply every term in the first lot of brackets by every term in the second lot of brackets, which can be done by following the steps below:

1.Write out each of the multiplications you need to perform, by putting each term in the first lot of brackets together with each term in the second lot of brackets and adding these terms together.
For example, when expanding
(a - 3)(a + 4) write out (a)(a) + (a)(4) + (-3)(a) + (-3)(4)

2.Perform each of the required multiplications, leaving each result in brackets in order to keep track of any negative signs. These will be dealt with in the next step.
For example, multiplying pairs of terms in
(a)(a) + (a)(4) + (-3)(a) + (-3)(4) gives (a2) + (4a) + (-3a) + (-12)

3.Remove the brackets from each term, replacing a positive sign with a negative sign if there is one inside the corresponding set of brackets.
For example, removing brackets and using appropriate signs for
(a2) + (4a) + (-3a) + (-12) gives a2 + 4a - 3a - 12

4.Simplify the expression by adding and/or subtracting like terms as required.
For example, simplifying
a2 + 4a - 3a - 12 gives a2 + a - 12

Examples: Expanding Brackets

1. Expand the brackets in the following algebraic equations:

2. Expand the pairs of brackets in the following algebraic equations:

a.b(3 + b) = (b)(3) + (b)(b)

= 3b + b2

b.3xy(6x - y 2 + x 2y) = (3xy)(6x) + (3xy)(-y 2) + (3xy)(x 2y)

= (18x 2y) + (-3xy 3) + (3x 3y 2)

= 18x 2y -3xy 2 + 3x3y 2

a.(e + f)(g + f) = (e)(g) + (e)(f) + (f)(g) + (f)(f)

= eg + ef + fg + f 2

b.(c - 4d)(-c - 3d) = (c)(-c) + (c)(-3d) + (-4d)(-c) + (-4d)(-3d)

= (-c 2) + (-3cd) + (4cd) + (12d 2)

= - c 2 - 3cd + 4cd + 12d 2

= - c 2 + cd + 12d 2

Activity 1: Practice Questions

Factorising with One Set of Brackets

Factorising is essentially the opposite of expanding brackets; instead of removing brackets, you put them into an algebraic expression.

One example of this is when you factorise an algebraic expression using one set of brackets. You can do this by following the two steps below:

1.Identify the highest common factor for all terms in the expression; that is, the largest term that divides into each term in the expression. This may be a constant or a variable or variables, or both.

For example to factorise 3x2 + 21x you should identify the highest common factor as 3x, since 3 divides into each term in the expression a whole number of times and there is one x in each term.

2.Put the highest common factor outside of a set of brackets, and put what remains of each term in the original expression inside the brackets.

For example once the highest common factor of 3x is taken out of the expression 3x2 + 21x the first term is reduced to x, while the second term is reduced to 7. Hence the factorised expression is 3x(x + 7)

1.Identify the highest common factor for all terms in the expression; that is, the largest term that divides into each term in the expression. This may be a constant or a variable or variables, or both.

For example to factorise 3x2 + 21x you should identify the highest common factor as 3x, since 3 divides into each term in the expression a whole number of times and there is one x in each term.

2.Put the highest common factor outside of a set of brackets, and put what remains of each term in the original expression inside the brackets.

For example once the highest common factor of 3x is taken out of the expression 3x2 + 21x the first term is reduced to x, while the second term is reduced to 7. Hence the factorised expression is 3x(x + 7)

Examples: Factorising with One Set of Brackets

Factorise the following algebraic expressions:

1. 14a + 8ab

Highest common factor is 2a

So factorised expression is 2a(7 + 4b)

2. 12x 3y - 3x 2y 2 + 6x 2yz

Highest common factor is 3x 2y

So factorised expression is 3x 2y(4x - y + 2z)

Highest common factor is 2a

So factorised expression is 2a(7 + 4b)

Highest common factor is 3x 2y

So factorised expression is 3x 2y(4x - y + 2z)

Activity 2: Practice Questions

Factorising with Two Sets of Brackets

Another example of factorising is when you factorise an algebraic expression using two sets of brackets. Generally this involves factorising a quadratic, which is an algebraic expression of the form:

ax 2 + bx + c, where a, b and c are numbers, and a ≠ 0

We will focus on factorising quadratics for which a = 1 in this topic. You can do this by following the steps below:

1. Write out two sets of brackets, and put the variable used in the expression inside both brackets.
For example to factorise
x 2 + 7x + 12, write out (x )(x )

2. Determine pairs of factors for the last term in the expression, taking into account whether these factors are positive or negative.
For example to factorise x 2 + 7x + 12,
determine that the pairs of factors for 12 are 1 and 12, 2 and 6 and 3 and 4.

3. Determine which of these pairs of factors can be added together or subtracted from one another to give the constant associated with the middle term in the expression. Then write one of these in each of the sets of brackets, being sure to use appropriate signs.
For example to factorise x 2 + 7x + 12, determine that the factors that can be added together to make 7 are 3 and 4. Hence the factorised form of the expression
is (x + 3)(x + 4)

Examples: Factorising with Two Set of Brackets

Factorise the following algebraic expressions:

1. x2+ x - 12

2. y2 - 8y + 12

3. z2 - 4z - 12

Factorisation will be of the form (x)(x)

Pairs of factors of 12 are 1 and 12, 2 and 6 and 3 and 4; one in each pair positive and one negative

Factors we require to give 1 are 4 and -3, since 4 - 3 = 1

Hence factorisation is (x+ 4)(x- 3)

Factorisation will be of the form (y)(y)

Pairs of factors of 12 are 1 and 12, 2 and 6 and 3 and 4; both numbers in each pair of factors either positive or negative

Factors we require to give -8 are -2 and -6, since -2 - 6 = -8

Hence factorisation is (y - 2)(y - 6)

Factorisation will be of the form (z)(z)

Pairs of factors of 12 are 1 and 12, 2 and 6 and 3 and 4; one in each pair positive and one negative

Factors we require to give -4 are 2 and -6, since 2 - 6 = -4

Hence factorisation is (z + 2)(z - 6)

End of Topic

Congratulations, you have completed this topic.

You should now have a better understanding of expanding brackets and factorising.

 


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