Factorising is an important process involved in many areas of mathematics. It is the process of rewriting an algebraic expression as a product of its factors and is used to solve equations, investigate the nature of mathematical relationships and many other applications. We will review factorisation methods below with a focus on those involving quadratic expressions.
HCF factorisation
This method can be applied to an expression with any number of terms that share common factors. The key is to take out the highest common factor:
| $AB+AC+\dots$AB+AC+… | $=$= | $A\left(B+C+\dots\right)$A(B+C+…) |
Difference of two squares
To use this factorisation, look for the difference of two terms which are both perfect squares.
| $A^2-B^2$A2−B2 | $=$= | $\left(A+B\right)\left(A-B\right)$(A+B)(A−B) |
For example:
| $16-y^2$16−y2 | $=$= | $(4+y)(4-y)$(4+y)(4−y) |
Grouping in pairs
Look for four terms which can be split into two pairs and factorised separately. Then factorise the resulting pair of terms afterwards. For example:
| $2x+4+xy+2y$2x+4+xy+2y | $=$= | $2\left(x+2\right)+y\left(x+2\right)$2(x+2)+y(x+2) |
| $=$= | $\left(x+2\right)\left(2+y\right)$(x+2)(2+y) |
Perfect squares
Look for three terms where the squared and constant terms are perfect squares, and the other term is twice the product of their square roots.
| $A^2+2AB+B^2$A2+2AB+B2 | $=$= | $\left(A+B\right)^2$(A+B)2 |
For example:
| $9x^2+12x+4$9x2+12x+4 | $=$= | $\left(3x+2\right)^2$(3x+2)2 |
Monic quadratics
To factorise $x^2+Px+Q$x2+Px+Q, where $P$P and $Q$Q are any numbers and the coefficient of $x^2$x2 is $1$1, first check if it is a perfect square. Otherwise, we look to factorise by finding two values $a$a and $b$b such that $a+b=P$a+b=P and $a\times b=Q$a×b=Q. We can then factorise the quadratic into the form $(x+a)(x+b)$(x+a)(x+b).
| $x^2+Px+Q$x2+Px+Q | $=$= | $(x+a)(x+b)$(x+a)(x+b) |
where $a+b=P$a+b=P and $a\times b=Q$a×b=Q
For example:
| $x^2+5x+6$x2+5x+6 | $=$= | $\left(x+3\right)\left(x+2\right)$(x+3)(x+2) |
Non-monic quadratics
Similar to monic quadratics, but the coefficient of $x^2$x2 is not $1$1. To factorise, first check for HCF factorisation or perfect square factorisation. Otherwise, factorise by using either the Product Sum Factors method or the Cross method.
Cross method
Let's have a look at factorising $5x^2+11x-12$5x2+11x−12 using the cross method. We must draw a cross with a possible pair of factors of $5x^2$5x2 on one side and another possible factor pair of $-12$−12 on the other side.
Let's start with the factor pairs of $5x$5x & $x$x on the left, and $-6$−6 & $2$2 on the other:
$5x\times2+x\times\left(-6\right)=4x$5x×2+x×(−6)=4x, which is incorrect, so let's try again with another two pairs:
$5x\times3+x\times\left(-4\right)=11x$5x×3+x×(−4)=11x which is the right answer. By reading across in the two circles, the quadratic must then factorise to $\left(5x-4\right)\left(x+3\right)$(5x−4)(x+3).
PSF method
The PSF (Product, Sum, Factor) method uses a similar idea we had with monic quadratics where we think about sums and products, but slightly differently.
For a quadratic in the form $ax^2+bx+c$ax2+bx+c:
1. Find two numbers, $m$m & $n$n, that have a SUM of $b$b and a PRODUCT of $ac$ac.
2. Rewrite the quadratic as $ax^2+mx+nx+c$ax2+mx+nx+c.
3. Use grouping in pairs to factorise the four-termed expression.
Worked example
example 1
Using the same example as above, factorise $5x^2+11x-12$5x2+11x−12 using the PSF method.
Think about what the sum and product of $m$m & $n$n should be
Do
We want the sum of of $m$m & $n$n to be $11$11, and the product to be $5\times\left(-12\right)=-60$5×(−12)=−60.
The two numbers work out to be $-4$−4 & $15$15, so:
| $5x^2+11x-12$5x2+11x−12 | $=$= | $5x^2-4x+15x-12$5x2−4x+15x−12 |
| $=$= | $x\left(5x-4\right)+3\left(5x-4\right)$x(5x−4)+3(5x−4) | |
| $=$= | $\left(5x-4\right)\left(x+3\right)$(5x−4)(x+3) |
This is the same answer that we got before!
Before looking for more complicated factorisations (such as for non-monic quadratics), it is a good idea to carefully check for any common factors first.
Also, after factorising an expression, make sure to check and see if any further factorisation can be done!
Practice questions
Question 1
Factorise the expression $\left(y+4\right)\left(y+7\right)+x\left(y+7\right)$(y+4)(y+7)+x(y+7).
Question 2
Factorise the expression $\left(x+13\right)^2-y^2$(x+13)2−y2.
Question 3
Factorise the expression $36-12u+u^2$36−12u+u2.
Question 4
Factorise the expression $20x^2-25x-30$20x2−25x−30.