Expanding brackets

Expanding brackets means multiplying everything inside the bracket by the letter or number outside the bracket. For example, in the expression 3(m + 7) both m and 7 must be multiplied by 3:

3(m + 7) = 3 \times m + 3 \times 7 = 3m + 21.

Expanding brackets involves using the skills of simplifying algebra. Remember that 2 \times a = 2a and a \times a = a^2.

Example

Expand 4(3n + y).

4(3n + y) = 4 \times 3n + 4 \times y = 12n + 4y

Question

Expand k(k - 2).

k(k - 2) = k \times k - 2 \times k = k^2 - 2k

Question

Expand 3f(5 - 6f).

3f(5 - 6f) = 3f \times 5 - 3f \times 6f = 15f - 18f^2

Expanding brackets with powers

Powers or index numbers are the floating numbers next to terms that show how many times a letter or number has been multiplied by itself. For example, a^2 = a \times a and a^4 = a \times a \times a \times a.

Using index laws, terms that contain powers can be simplified. Remember that multiplying indices means adding the powers. For example, a^2 \times a^3 = (a \times a) \times (a \times a \times a) = a^5.

Example

Expand the bracket 3b^2(2b^3 + 3b).

Multiply 3b^2 by 2b^3 first. 3 \times 2 = 6 and b^2 \times b^3 = b^5, so 3b^2 \times 2b^3 = 6b^5.

Then multiply 3b^2 by 3b. 3 \times 3 = 9 and b^2 \times b = b^3, so 3b^2 \times 3b = 9b^3.

So, 3b^2(2b^3 + 3b) = 6b^5 + 9b^3.

Question

Expand the bracket 5p^3q(4pq - 2p^5q^2 + 3p).

Multiply 5p^3q by 4pq. 5p^3q \times 4pq = 20p^4q^2.

Multiply 5p^3q by -2p^5q^2. 5p^3q \times - 2p^5q^2 = - 10p^8q^3.

Multiply 5p^3q by 3p. 5p^3q \times 3p = 15p^4q.

5p^3q(4pq - 2p^5q^2 + 3p) = 20p^4q^2 - 10p^8q^3 + 15p^4q

Expanding and simplifying

Expressions with brackets can often be mixed in with other terms. For example, 3(h + 2) - 4. In these cases first expand the bracket and then collect any like terms.

Example 1

Expand and simplify 3(h + 2) - 4.

3(h + 2) - 4 = 3 \times h + 3 \times 2 - 4 = 3h + 6 - 4 = 3h + 2

Example 2

Expand and simplify 6g + 2g(3g + 7).

BIDMAS or BODMAS is the order of operations: Brackets, Indices or Powers, Divide or Multiply, Add or Subtract.

Following BIDMAS, multiplying out the bracket must happen before completing the addition, so multiply out the bracket first.

This gives: 6g + 2g(3g + 7) = 6g + 2g \times 3g + 2g \times 7 = 6g + 6g^2 + 14g

Collecting the like terms gives 6g^2 + 20g.

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Answers are usually written with descending order of powers.