Expanding and factorising brackets

To expand a bracket, multiply every term inside by the term outside.

1. Multiply the term outside by the first term inside.
2. Multiply it by each other term inside, keeping the signs.
3. Collect like terms if there are any.

Example: expand 3(2x − 5) and x(x + 4).

1. 3 × 2x = 6x and 3 × −5 = −15, so 6x − 15
2. x × x = x² and x × 4 = 4x, so x² + 4x

Harder example: expand and simplify 4(x + 2) − 3(x − 5).

1. 4(x + 2) = 4x + 8
2. −3(x − 5) = −3x + 15, because −3 × −5 = +15
3. 4x + 8 − 3x + 15 = x + 23

Multiply each term in the first bracket by each term in the second. FOIL helps you get all four: First, Outside, Inside, Last.

Example: expand (x + 3)(x − 5).

1. First: x × x = x²
2. Outside: x × −5 = −5x
3. Inside: 3 × x = 3x
4. Last: 3 × −5 = −15
5. x² − 5x + 3x − 15 = x² − 2x − 15

The grid method gives the same four terms:

Example: expand (x + 3)².

1. (x + 3)² = (x + 3)(x + 3)
2. x² + 3x + 3x + 9 = x² + 6x + 9

Expand two of the brackets first, then multiply the answer by the third.

1. Expand the first two brackets and simplify.
2. Multiply every term of that answer by each term of the third bracket.
3. Collect like terms.

Example: expand (x + 1)(x + 2)(x − 3).

1. (x + 1)(x + 2) = x² + 2x + x + 2 = x² + 3x + 2
2. (x² + 3x + 2)(x − 3) = x³ − 3x² + 3x² − 9x + 2x − 6
3. x³ − 7x − 6

Check by putting in a value, such as x = 2: (3)(4)(−1) = −12, and 8 − 14 − 6 = −12.

To factorise is to put brackets back in: the opposite of expanding.

1. Find the highest common factor of the numbers.
2. Find any letters that are in every term (with their lowest power).
3. Write the HCF outside the bracket.
4. Divide each term by it to find what goes inside.
5. Check by expanding.

Example: factorise 6x + 15.

1. The HCF of 6 and 15 is 3, and x isn't in both terms.
2. 3(2x + 5)

Harder example: factorise 8x²y − 12xy².

1. HCF of 8 and 12: 4
2. Letters in both: x and y
3. 8x²y ÷ 4xy = 2x and 12xy² ÷ 4xy = 3y
4. 4xy(2x − 3y)