Module 1 (M1) - Numbers - Indices

Expanding and simplifying

Two linear expressions can be expanded, and then simplified by collecting like terms, if there are any.

Linear expressions have no terms with power of 2 or higher.

For example,
15x – 4y is a linear expression but 15x – 4y² is not.

Expanding and simplifying two linear expressions

Two linear expressions can be multiplied together by separating the terms in the first bracket and then multiplying each in turn by the terms in the second bracket.The expanded expression can then be simplified by collecting like terms.

Example

Expand the brackets and simplify fully \((p+ 3q)(3p – 5q)\)

Solution
This time there are terms in ‘p’ and terms in ‘q’ with no stand-alone numbers, but the method is exactly the same.

\(\mathbf{p(3p - 5q) + 3q(3p - 5q)} = {3p^2- 5qp} +9qp - 15q^2{+ 3x - 12}\)

Remember that p x -5q = -5pq or -5qp
The order of the letters does not matter but the number always comes first.

\(3p^2 {- 5qp + 9qp} - 15q^2 = 3p^2 + 4qp - 15q^2\)

Answer
\((p+ 3q)(3p – 5q) = 3p^2 + 4qp - 15q^2\)

Example

Expand and simplify \((2x - 3)(3x - 5)\)

Solution

Separate the ‘2x’ and the ‘-3’ from the first bracket and multiply each by (3x - 5)

\(\mathbf {(2x -3)(3x - 5) = 2x (3x - 5) + 3 (3x - 5)}\)

Expand the brackets

\(\mathbf {{2x (3x - 5)}{- 3 (3x - 5)} = {6x^2- 10x }{- 9x - 12}}\)

Be careful with signs when multiplying out the second bracket!

Now collect like terms

\(6x^2 {- 10x + 9x} +15 = 6x^2 {-19x} + 15 \)

Answer
\((2x - 3)(3x - 5) = 6x^2 – 19x + 15\)

Example

Expand the brackets and simplify fully \((p+ 3q)(3p – 5q)\)

Solution
This time there are terms in ‘p’ and terms in ‘q’ with no stand-alone numbers, but the method is exactly the same.

\(\mathbf {p(3p - 5q) {+ 3q(3p - 5q)} = {3p^2- 5qp} +9qp - 15q^2 {+ 3x - 12}}\)

Remember that p x -5q = -5pq or -5qp
The order of the letters does not matter but the number always comes first.

\(3p^2 {- 5qp + 9qp} - 15q^2 = 3p^2 + 4qp - 15q^2\)

Answer
\((p+ 3q)(3p – 5q) = 3p^2 + 4qp - 15q^2\)

Alternative FOIL method

FOIL Method (2x + 3)(5x - 1) = 10x2 - 2x + 15x - 3

An alternative method is the FOIL method. It can be quicker especially if you are confident with + and – numbers.

FOIL stands for

First - Multiply the first terms in the brackets
Outside - Multiply the outside terms
Inside - Multiply the inside terms
Last - Multiply the last terms

Now collect like terms

\(10x^2 {- 2x + 15x} - 3 = 10x^2 {+ 13x} - 3\)

Answer
\((x + 3)(x - 4) = 10x^2 + 13x – 3\)

Question

Expand the brackets and simplify this expression

(2y – 7)(4y + 1)

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