Coaching Centre
Factorising: common factors, difference of squares and grouping
Taking out the highest common factor, factorising the difference of two squares, and factorising four-term expressions by grouping in pairs.
Common factors
Always look for a common factor first: $6x^2 + 9x = 3x(2x + 3)$.
Difference of two squares
Difference of two squares
$$a^2 - b^2 = (a - b)(a + b)$$Factorise $4a^2 - 9$ and $2x^2 - 50$.
$4a^2 - 9 = (2a - 3)(2a + 3)$
$2x^2 - 50 = 2(x^2 - 25) = 2(x - 5)(x + 5)$
Grouping in pairs
Factorise $x^2 + 3x + 2x + 6$.
$x(x + 3) + 2(x + 3)$
$= (x + 3)(x + 2)$
$x^2 - 49$
$(x - 7)(x + 7)$
First step in any factorising
Take out the HCF.
$ax + ay + bx + by$
$(a + b)(x + y)$
Factorise $4a^2 - 9$.
$(2a)^2 - 3^2$.
Factorise $2x^2 - 50$ fully.
HCF $2$ first.
Factorise $x^2 + 3x + 2x + 6$.
$x(x + 3) + 2(x + 3)$.
Factorise $6x^2 + 9x$.
HCF $3x$.
Factorise $x^2 - 49$.
DOTS.
Factorise $ax + ay + bx + by$.
$a(x + y) + b(x + y)$.
Factorise $x^2 + 5x - 2x - 10$.
$x(x + 5) - 2(x + 5)$.
Which cannot be factorised as a difference of two squares?
It is a sum, not a difference.
New numbers every time. Keep going until you get five in a row.