Factorisation

Step 1: Find the HCF of the numbers.

• The factors of 16 are {1, 2, 4, 8, 16}.
• The factors of 24 are {1, 2, 3, 4, 6, 8, 12, 24}.
• The numerical HCF is $8$.

Step 2: Find the HCF of the variables.

• For the variable $m$, the powers are $m^3$ and $m^2$. The lowest power is $m^2$.
• For the variable $n$, the powers are $n^1$ and $n^2$. The lowest power is $n$.
• The variable HCF is $m^2n$.

Step 3: Combine to find the overall HCF.

• Overall HCF = $8m^2n$.

Step 4: Divide each term by the HCF to find the contents of the bracket.

• $16m^3n \div 8m^2n = 2m$
• $-24m^2n^2 \div 8m^2n = -3n$

Step 5: Write the final single-bracket expression.

$$8m^2n(2m - 3n)$$

Factorising Quadratics in the Form $x^2 + bx + c$

Working backwards to find which two numbers were multiplied together is a brilliant puzzle that unlocks quadratic equations. A monic quadratic is an expression where the leading coefficient (the number in front of the $x^2$) is exactly 1. When a quadratic expression contains three terms, it is known as a trinomial.

To factorise a monic quadratic trinomial, the result will take the form of double brackets, written as $(x + p)(x + q)$. To find the unknown values $p$ and $q$, we use the sum and product method. You must find factor pairs of the constant term ($c$) that multiply to give , but add together to give the coefficient of ().

The sign of the constant term $c$ gives you a huge clue:

• If $c$ is positive, both numbers in the brackets will share the same sign as $b$.
• If $c$ is negative, the numbers will have opposite signs, and the larger number will take the sign of $b$.

Factorise $x^2 - 5x - 24$

Step 1: Identify $b$ and $c$.

• $b = -5$
• $c = -24$

Step 2: List factor pairs of $c$ ($-24$) and check their sums.

• $1$ and $-24$ (Sum = $-23$)
• $2$ and $-12$ (Sum = $-10$)
• and (Sum = )

Step 3: Place these numbers into double brackets.

$$(x + 3)(x - 8)$$

Step 4: Verify by expanding mentally.

• $x(x) - 8(x) + 3(x) + 3(-8) = x^2 - 8x + 3x - 24 = x^2 - 5x - 24$.

Factorising Quadratics in the Form $ax^2 + bx + c$

When the $x^2$ term has a number in front of it, the simple sum-and-product rule isn't enough; we need a more structured method to break the expression apart. A non-monic quadratic is a quadratic where the leading coefficient ($a$) is not 1. To factorise these, we use a systematic process called the ac method.

First, multiply the leading coefficient ($a$) by the constant term ($c$) to find the "master product". You then find two numbers that multiply to this $ac$ value and add to give the middle coefficient ($b$). These numbers are used for splitting the middle term, rewriting the original three-term trinomial into four terms.

Once you have four terms, you can use factorising by grouping. You split the expression down the middle, take a common factor from the first pair, and a common factor from the second pair. This will reveal an identical bracket in both parts, which becomes one of your final double brackets.

Factorise $3x^2 - 14x + 8$

Step 1: Identify $a$, $b$, and $c$, then calculate $ac$.

• $a = 3$, $b = -14$, $c = 8$
• $ac = 3 \times 8 = 24$

Step 2: Find factors of $24$ that sum to $-14$.

• The factors are $-12$ and $-2$ ($-12 \times -2 = 24$, and $-12 + -2 = -14$).

Step 3: Split the middle term ($-14x$) using these factors.

• $3x^2 - 12x - 2x + 8$

Step 4: Factorise by grouping the pairs.

• Group 1: $3x^2 - 12x$ factorises to $3x(x - 4)$
• Group 2: $-2x + 8$ factorises to

Step 5: Write the final factorised expression by combining the outside terms into one bracket, and the common bracket into the other.

$$(3x - 2)(x - 4)$$