Factorising What is factorising It should be obvious
Factorising
What is factorising? It should be obvious, that factorising is just the reverse process of expanding! It means writing an expression as a product of its factors. We just use the rules of expanding in reverse (using brackets). We look to take out the highest common factor (HCF) of the terms in the expression. In effect, we are “un-distributing” the expression back into bracket form.
What is a factor? A factor is simply a term that divides into an expression or number without any remainder. Factors of 12 are 1, 2, 3, 4, 6, 12 Factors of a 2 b are a, a 2, b Factors of 12 a 2 b are the many different combinations of above i. e. 2 a, 6 b, 3 a 2, 4 a 2 b The highest common factor (HCF) is the largest factor that is shared by all the terms.
Common Factors The common factor is a term which is common to elements of the expression. We always identify the highest common factor. It could be a number, variable, or combination of both. Example: For the expression 12 ab + 8 b The highest common factor of 12 ab and 8 b is 4 b • • 4 is the highest factor of 12 and 8 b is the only common factor of ab and b Therefore we can write 12 ab + 8 b as 4 b(3 a + 2)
The distributive law (revisited) The reverse of the distribution law previously learnt is ab + ac = a(b + c) The expression ab + ac can be expressed as the product of the two factors, ‘a’ and ‘(b + c)’. Here ‘a’ is common to both terms, and can be removed as the common factor, leaving ‘b’ and ‘c’ behind. The sign between ‘b’ and ‘c’ is decided by the sign between the terms from the 1 st expression.
Example Factorise: 12 xy – 8 yz = 4 ⨯ 3 ⨯ x ⨯ y ‒ 4 ⨯ 2 ⨯ y ⨯ z 4 and y are distributed (common), so come out the front as the common factors to give us 12 xy – 8 yz = 4 y(3 x – 2 z) ANSWER
Example Factorise 3 x 2 + 4 x = 3 ⨯ x ⧾ 4 ⨯ x There is no number common factor, only the variable x 3 x 2 + 4 x = x(3 x + 4) ANSWER
Practice Factorise the following: 1. 4 a + 12 2. xy – 3 x 3. 6 x + 8 y 4. ab – a 2 5. 3 p 2 – 6 p 6. 3 x 3 + 6 x 2 – 3 x 7. 8 x – 24 y + 20 8. 3 a 2 b + 5 a 2 b + ab 9. 3 x + 6 y + ax + 2 ay
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