Factorisation
ICSE Class 9 · Mathematics · Chapter 5
What this chapter covers
- Factors, factorisation and the common factor
- The difference of two squares: a2 - b2
- Harder forms of the difference of two squares
- The sum and difference of two cubes: a3 ± b3
- Harder forms with cubes
- Splitting the middle term: trinomials of the form x2 + bx + c
- Splitting the middle term when a ≠ 1
- Trinomials in two variables and in disguise
- Choosing the method and factorising completely
- Common mistakes and how to avoid them
Key points
- Factorisation writes an expression as a product of factors; it is the reverse of expansion.
- Always take out the highest common factor first, and always factorise completely.
- Difference of two squares: a2 - b2 = (a + b)(a - b); a simple sum of two squares such as x2 + 9y2 does not factorise at this level, though a sum of higher powers such as x4 + 4 may yield to adding and subtracting the same term.
- Hidden squares appear in higher even powers, in perfect square trinomials inside longer expressions, and after adding and subtracting a term.
- Sum of two cubes: a3 + b3 = (a + b)(a2 - ab + b2).
- Difference of two cubes: a3 - b3 = (a - b)(a2 + ab + b2); remember SOAP for the signs.
- When an expression is both a difference of squares and a difference of cubes, use squares first.
- To factorise ax2 + bx + c, find two numbers with product ac and sum b, split the middle term, then group in pairs.
- If no factor pair of ac has sum b, the trinomial does not factorise with integer coefficients.
- A repeated bracket can be replaced by one letter while factorising, but the final answer must be in the original variables.
- Check every answer by multiplying the factors back out.
Questions answered
Define factorisation. How is it related to expansion?
Factorisation is the process of writing an algebraic expression as a product of two or more factors. Expansion multiplies the factors back out, so each process undoes the other.
Factorise x4 + 4.
Add and subtract 4x2: x4 + 4x2 + 4 - 4x2 = (x2 + 2)2 - (2x)2 = (x2 + 2x + 2)(x2 - 2x + 2).
Factorise 5x3 - 45x completely.
5x(x2 - 9) = 5x(x + 3)(x - 3).
Factorise x4 - 1 completely.
(x2 + 1)(x + 1)(x - 1). The factor x2 + 1 is a sum of two squares and stays as it is.
Factorise 9a2 - 6a + 1 - 16b2.
The first three terms are (3a - 1)2, so the expression is (3a - 1)2 - (4b)2 = (3a - 1 + 4b)(3a - 1 - 4b).
Use factorisation to evaluate 572 - 432.
(57 + 43)(57 - 43) = 100 × 14 = 1400.
Factorise (x + 1)3 + 27.
With a = x + 1 and b = 3, the expression is (x + 1 + 3)[(x + 1)2 - 3(x + 1) + 9] = (x + 4)(x2 + 2x + 1 - 3x - 3 + 9) = (x + 4)(x2 - x + 7).
Factorise 27a3 - (1)/(8).
(3a - (1)/(2))(9a2 + (3a)/(2) + (1)/(4)).
Past board exam questions
ICSE 2019 sample paper · 3 marks
Factorize the given expression completely: 6x2+7x-5
Show answer
6x2+7x-5 = 6x2+10x-3x-5
= 2x(3x+5)-1(3x+5)
= (3x+5)(2x-1)
ICSE 2019 sample paper · 3 marks
Factorize: 4a2 - 9b2 - 16c2 + 24bc
Show answer
4a2 - (9b2 - 24bc + 16c2)
= (2a)2 - (3b - 4c)2
= (2a - 3b + 4c)(2a + 3b - 4c)