Year 10AC9M10A01

Expanding and Factorising

Expand, factorise and simplify expressions and solve equations algebraically, applying exponent laws involving products, quotients and powers of variables, and the distributive property.

Expanding and factorising are inverses. Recognising the special forms on sight saves most of the work.

Method

Binomial products

(a+b)(c+d) = ac + ad + bc + bd

Every term times every term.

Perfect square

(a+b)^2 = a^2 + 2ab + b^2

The middle term is the one people forget.

Difference of two squares

a^2 - b^2 = (a-b)(a+b)

Look for it whenever there is a minus and two squares.

Non-monic trinomials: split the middle

ax^2+bx+c

Find two numbers multiplying to $ac$ and adding to $b$.

Always take out a common factor first

It makes everything after it easier.

Worked example

e.g.

\text{Factorise } 6x^2 + 11x - 10

Check for a common factor. There is none.
Multiply $a$ by $c$ . $6 \times (-10) = -60$
Find two numbers multiplying to $-60$ and adding to 11. $15 \text{ and } -4$
Split the middle term using them. $6x^2 + 15x - 4x - 10$
Factorise in pairs. $3x(2x+5) - 2(2x+5)$
Take out the common bracket. $(2x+5)(3x-2)$

Practice

Common factor first. Then look for a special form before working.

(x+3)(x+5)

Answer

x^2 + 8x + 15

(2x-1)^2

Answer

4x^2 - 4x + 1

\text{Factorise } x^2 - 49

Answer

(x-7)(x+7)

\text{Factorise } 3x^2 + 10x + 8

Answer

(3x+4)(x+2)

\text{Simplify } \frac{x^2-9}{x+3}

Answer

x - 3

\text{Why check for a common factor first?}

Answer

\text{It leaves smaller numbers to work with.}