01.04 Factorising
In this lesson you will learn how to:
- Factorise expressions for which all terms have a common variable
- Factorise quadratic expressions in one variable
- Recognise difference of two squares
- Factorise quadratic expressions in two variables
- Factorise cubics with no constant term
Prerequisites from previous chapters:
- N/A
VIDEO LESSON
INTERACTIVE SELF-STUDY
In the previous lesson on Expanding Brackets we looked at how to expand products of expressions. Expanding an expression involves multiplying the factors together.
Factorising is the process of doing the opposite: breaking down an expression into its factors.
If the terms of an expansion have no common variables, then at best, you can only factor out a constant
As a general procedure, factor out the highest common factor of all coefficients. This will be the factor in front of the brackets
Then divide each term in the expression by the highest common factor. This will be the factor inside the brackets
Worked Example 1 – (GRADE E)
Factorise \(15x^2+20y\)
There are no variables common to all terms.
The highest common factor of \(15\) and \(20\) is \(5\)
Dividing each term by \(5\) gives us: \(\phantom{-}\) \(3x^2\) and \(4y\).
Hence \(15x^2+20y ≡ 5(3x^2+4y)\)
Example 2 – (GRADE E)
Factorise the following expressions
a. \(4x^2-16y\)
Factorising Expressions Containing Common Variables
We can factorise expressions for which all terms have a common variable.
A general procedure is to factor out the lowest power present of each variable.
You should also factor out the highest common factor of all coefficients.
Then divide each term in the expanded form by the quantity you have factored out. This will be the factor inside the brackets
Worked Example 3 – (GRADE E)
Factorise \(12x^2 y^5-42x^3 y^3\)
All terms contain \(x\) and \(y\).
The lowest power of \(x\) present is \(x^2\).
The lowest power of \(y\) present is \(y^3\).
The highest common factor of \(12\) and \(42\) is \(6\)
Hence we can factor out \(6x^2y^3\)
Now we can divide each term in the expression by \(6x^2y^3\): \(\phantom{-}\) \(2y^2\) and \(-7x\).
This tells us \(12x^2 y^5-42x^3 y^3≡6x^2y^3(2y^2-7x)\)
Example 4 – (GRADE E)
Factorise the following expressions
a. \(5x^2+10xy\)
c. \(4x^3y^2-18xy^4\)
d. \(9x^2y-6xy^2\)
Sometimes the coefficients may not be integers and are instead rational numbers (fractions)
You can factor out \(\displaystyle \frac{p}{q}\) where:
- The numerator \(p\) is the highest common factor of the numerators of the coefficients
- The denominator \(q\) is the lowest common multiple of the denominators of the coefficients
If one of the coefficients is an integer, rewrite it as a fraction, e.g. write \(6\) as \(\displaystyle \frac{6}{1}\)
Worked Example 5 – (GRADE D)
Factorise \(\displaystyle \frac{3}{8}x^3-\frac{7}{12}x^2\)
\(x\) is present in every term and the lowest power present is \(x^2\)
The highest common factor of the numerators (\(3\) and \(7\)) is \(1\)
The lowest common multiple of the denominators (\(8\) and \(12\)) is \(24\)
Hence we can factor out \(\displaystyle \frac{1}{24}x^2\) from each term.
Then \(\displaystyle \frac{3}{8}x^3-\frac{7}{12}x^2 ≡\frac{1}{24}x^2(9x-14)\)
Example 6 – (GRADE D)
Factorise the following expressions
a. \(\displaystyle \frac{3}{2}x^2y^2+\frac{9}{4}xy^2\)
c. \(\displaystyle \frac{5}{3}x^4-8x^3\)
FACTORISING QUADRATIC EXPRESSIONS IN ONE VARIABLE
A quadratic expression in one variable has the form \(ax^2+bx+c\), where \(a,b\), and \(c\) are real numbers and \(a≠0\)
- The real numbers are all positive and negative numbers, or zero, including fractions, decimals and surds
We will first consider quadratics where the coefficient of \(x^2\) is \(a=1\). (These are called monic quadratics)
Then the quadratics will be of the form \(x^2+bx+c\)
i. Expand and simplify \((x+m)(x+n)\), where \(m\) and \(n\) are constants.
\((x+m)(x+n)=x^2+(m+n)x+mn\).
If we compare this with \(x^2+bx+c\), we can see that \(m+n=b\) and \(mn=c\).
- Hence to factorise \(x^2+bx+c\) in the form \((x+m)(x+n)\), we need to find two numbers \(m\) and \(n\) such that they add to \(b\) and multiply to \(c\).
It is easiest to first look for pairs of factors of \(c\). This includes both positive and negative factors.
Example 7 – (GRADE E)
Factorise the following quadratics
a. \(x^2+5x+6\)
c. \(x^2-4x-12\)
Now let’s focus on quadratic expressions where the coefficient of \(x^2\) is not equal to \(1\).
As a general procedure
- Multiply \(a\) and \(c\)
- Find two factors of \(ac\) that add up to \(b\)
- Rewrite the \(b\) term as the sum of these two factors
- Factorise each pair of terms
- Take out the common factor
Worked Example 8 – (GRADE D)
Factorise \(10x^2-13x-3\)
Step 1: Find \(ac\)
\(ac=-30\)
Step 2: Find pairs of factors of \(ac\) that sum to \(b\)
Pairs of factors of \(-30\) are \((±1, \mp 30)\), \((±2, \mp 15)\), \((±3, \mp 10)\), \((±5, \mp 6)\).
The pair that adds to \(-13\) is \((2, -15)\)
Step 3: Rewrite the \(b\) term as a sum of the pair of factors
\(10x^2+2x-15x-3\)
Step 4: Factorise each pair of terms
\(\underbrace{\color{blue}{10x^2+2x}}_{\text{factorise}} \phantom{-} \underbrace{\color{red}{-15x-3}}_{\text{factorise}}\)
\(=2x(5x+1) -3(5x+1)\)
Step 5: Take out the common factor
\((5x+1)(2x-3)\)
- \((5x+1)\) and \((2x-3)\) are called linear factors of \(10x^2-13x-3\)
- a linear expression in \(x\) is of the form \(ax+b\), where \(a\) and \(b\) are constants
Example 9 – (GRADE D)
Factorise the following quadratics
a. \(12x^2+17x+6\)
i. Expand and simplify \((a+b)(a-b)\)
- \(a^2-b^2≡(a+b)(a-b)\).
- \(a^2-b^2\) is called a difference of two squares.
Differences of two squares will appear all the time in A-Level Maths so it is important to be able to spot and factorise them
- \(a^2-b^2\) is called a difference of two squares.
Example 10 – (GRADE D)
Factorise the following quadratic expressions
a. \(x^2-9\)
c. \(x^2-5\)
- For some quadratic expressions, all the coefficients may have a common factor. If you spot this, factor out the highest common factor from the entire expression to make it easier to factorise.
- If the \(x^2\) coefficient is negative, factor out \(-1\) from the entire quadratic expression.
- If some of the coefficients are fractions, first factor out \(\displaystyle \frac{p}{q}\) from the entire quadratic expression, where \(p\) is the highest common factor of all numerators and \(q\) is the lowest common multiple of all denominators.
Example 11 – (GRADE D)
Factorise the following expressions
a. \(3x^2 +36x+108\)
c. \(-14x^2+35x-21\)
d. \(-9x^2+36\)
e. \(\displaystyle \frac{6}{5}x^2+9x-\frac{24}{5}\)
FACTORISING QUADRATIC EXPRESSIONS IN TWO VARIABLES
i. Expand \((x+y)(x+2y)\)
So we know \(x^2+3xy+2y^2\) factorises to \((x+y)(x+2y)\)
But how can we factorise a general quadratic expression in two variables of the form \(ax^2+bxy+cy^2\)?
As a general procedure, we can let \(y=1\) and treat this as a quadratic expression in one variable.
We can then factorise new quadratic expression as we usually would into the form \((px+q)(rx+s)\)
Finally we put \(y\) back into the factors: \((px+qy)(rx+sy)\)
Worked Example 12 – (GRADE C)
Factorise \(6x^2+5xy-4y^2\)
Step 1: Let \(y=1\)
Then we obtain \(6x^2+5x-4\)
Step 2: Factorise this quadratic
\(ac=-24\)
The pair of factors of \(-24\) that sum to \(5\) is \((8, -3)\)
Hence \(6x^2+5x-4=6x^2+8x-3x-4\)
\(=2x(3x+4) -1(3x+4)\)
\(=(3x+4)(2x-1)\)
Step 3: Place a \(y\) next to the constants
Hence \(6x^2+5xy-4y^2≡(3x+4y)(2x-y)\)
Example 13 – (GRADE C)
Factorise the following expressions
a. \(5x^2-7xy-6y^2\)
FACTORISING MORE COMPLEX EXPRESSIONS
You may need to use multiple factorisation techniques to completely factorise more complicated expressions
As a general procedure:
- Factor out the highest common factor of all coefficients or an appropriate fraction \(\displaystyle \frac{p}{q}\)
- If all terms have common variables, factor out the lowest power present of each variable
- For whatever remains, see whether you can factorise a quadratic expression or recognise it as a difference of two squares
Example 14 – (GRADE C)
Fully factorise the following expressions
a. \(2x^4+14x^3+24x^2\)
c. \(x^4-81\)
d. \(x^3y^2+8x^2y^3+15xy^4\)
END OF LESSON!
You can now answer:
EDEXCEL Pure Year 1 Ex 1C – All Questions
OCR A Student Book 1 Ex 2 –