Method 1: Factorising "by inspection"
Method 2: Factorising "by grouping"
Method 3: Factorising "by using a grid"
x^{2} | -4x | |
+2x | -8 |
x | x^{2} | -4x |
+2x | -8 |
x | -4 | |
x | x^{2} | -4x |
+2x | -8 |
x | -4 | |
x | x^{2} | -4x |
+2 | +2x | -8 |
(a) Factorise .
We will factorise by inspection.
We need two numbers that:
multiply to -21, and sum to -4
-7, and +3 satisfy this
Write down the brackets.
(x + 3)(x - 7)
(b) Factorise .
We will factorise by splitting the middle term and grouping.
We need two numbers that:
multiply to 6, and sum to -5
-3, and -2 satisfy this
Split the middle term.
x^{2} - 2x - 3x + 6
Factorise x out of the first two terms.
x(x - 2) - 3x +6
Factorise -3 out of the last two terms.
x(x - 2) - 3(x - 2)
These have a common factor of (x - 2) which can be factored out.
(x - 2)(x - 3)
(c) Factorise .
We will factorise by using a grid.
We need two numbers that:
multiply to -24, and sum to -2
+4, and -6 satisfy this
Use these to split the -2x term and write in a grid.
x^{2} | +4x | |
-6x | -24 |
Write a heading using a common factor for the first row:
x | x^{2} | +4x |
-6x | -24 |
Work out the headings for the rows, e.g. “What does x need to be multiplied by to make x^{2}?”
x | +4 | |
x | x^{2} | +4x |
-6x | -24 |
Repeat for the heading for the remaining row, e.g. “What does x need to be multiplied by to make -6x?”
x | +4 | |
x | x^{2} | +4x |
-6 | -6x | -24 |
Read-off the factors from the column and row headings.
(x + 4)(x - 6)
Factorising a ≠ 1 "by grouping"
Factorising a ≠ 1 "by using a grid"
4x^{2} | -28x | |
+3x | -21 |
4x | 4x^{2} | -28x |
+3x | -21 |
x | -7 | |
4x | 4x^{2} | -28x |
+3x | -21 |
x | -7 | |
4x | 4x^{2} | -28x |
+3 | +3x | -21 |
As a check, expand your answer and make sure you get the same expression as the one you were trying to factorise.
(a) Factorise .
We will factorise by splitting the middle term and grouping.
We need two numbers that:
multiply to 6 × -3 = -18, and sum to -7
-9, and +2 satisfy this
Split the middle term.
6x^{2} + 2x - 9x - 3
Factorise 2x out of the first two terms.
2x(3x + 1) - 9x - 3
Factorise -3 of out the last two terms.
2x(3x + 1) - 3(3x + 1)
These have a common factor of (3x + 1) which can be factored out.
(3x + 1)(2x - 3)
(b) Factorise .
We will factorise by using a grid.
We need two numbers that:
multiply to 10 × -7 = -70, and sum to +9
-5, and +14 satisfy this
Use these to split the 9x term and write in a grid.
10x^{2} | -5x | |
+14x | -7 |
Write a heading using a common factor for the first row:
5x | 10x^{2} | -5x |
+14x | -7 |
Work out the headings for the rows, e.g. “What does 5x need to be multiplied by to make 10x^{2}?”
2x | -1 | |
5x | 10x^{2} | -5x |
+14x | -7 |
Repeat for the heading for the remaining row, e.g. “What does 2x need to be multiplied by to make +14x?”
2x | -1 | |
5x | 10x^{2} | -5x |
+7 | +14x | -7 |
Read-off the factors from the column and row headings.
(2x - 1)(5x + 7)
Recognise that and are both squared terms and the second term is subtracted from the first term - you can factorise using the difference of two squares.
Rewrite the expression with the square root of each term added together in the first bracket and subtracted from each other in the second bracket.
Recognise that and are both squared terms and the second term is subtracted from the first term - you can factorise using the difference of two squares.
Rewrite the expression with the square root of each term added together in the first bracket and subtracted from each other in the second bracket.