Factorisation
ICSE Class 8 · Mathematics · Chapter 13
What this chapter covers
- Factors and the idea of factorisation
- The highest common factor of terms
- Factorising by taking out a common factor: a(x + y)
- Factorising perfect squares: (x + y)² and (x − y)²
- Factorising the difference of two squares
- Factorising expressions of the form x² + (a + b)x + ab
- Choosing the right method
- Common errors in factorisation
Key points
- Factorisation writes an expression as a product of factors; it is the reverse of multiplication.
- The HCF of terms uses the HCF of the numbers and each common letter to its smallest power.
- Always take out the common factor first: ax + ay = a(x + y).
- A bracket shared by every term can be taken out like a single letter.
- x2 + 2xy + y2 = (x + y)2 and x2 - 2xy + y2 = (x - y)2; the middle term must be twice the product of the square roots.
- a2 - b2 = (a + b)(a - b); a sum of two squares does not factorise this way.
- For x2 + (a + b)x + ab, find two numbers whose product is the last term and whose sum is the middle coefficient.
- Keep factorising until no factor can be broken down further.
- Check every answer by multiplying out or by putting a small number in place of the letter.
Questions answered
Find the HCF of 18a2b and 24ab2.
The HCF of 18 and 24 is 6, and the smallest powers are a and b, so the HCF is 6ab.
Factorise 12x - 18y.
The HCF is 6, so 12x - 18y = 6(2x - 3y).
Factorise 7p2 + 7p.
The HCF is 7p, so 7p2 + 7p = 7p(p + 1).
Factorise m(x - 2) + 4(x - 2).
The common bracket is (x - 2), so the answer is (x - 2)(m + 4).
Factorise x2 + 14x + 49.
It is a perfect square, since 14x = 2 × x × 7, so the answer is (x + 7)2.
Factorise 4y2 - 12y + 9.
The first term is (2y)2, the last is 32, and 2 × 2y × 3 = 12y with a minus sign, so the answer is (2y - 3)2.
Factorise 81 - z2.
It is a difference of squares, 92 - z2, so the answer is (9 + z)(9 - z).
Use factorisation to find 1022 - 982.
(102 + 98)(102 - 98) = 200 × 4 = 800.