Factorising Quickly: Speed Up Your Algebra
Simplify algebra in seconds by spotting common factors.
Factorising is the reverse of expanding brackets. It is a superpower in algebra that allows you to simplify messy expressions quickly. To do it fast, you just need to spot the “Highest Common Factor” (HCF).
1. Spotting the Common Factor 🎯
Factorising means rewriting an expression as a product using brackets. Think of it as taking the expression apart to see what it is made of.
The Golden Rule: Look for the biggest number and the common letters that go into every part of the expression.
Let’s look at the expression:
$$10x + 15$$
To factorise this quickly, ask yourself two questions:
- Numbers: What is the biggest number that divides into 10 and 15? (Answer: 5)
- Letters: Is there a letter in both terms? (Answer: No, only the first term has an $x$)
So, the common factor is just $5$.
2. The Two-Step Method âš¡
Once you have the common factor, you can write the answer in seconds using this method.
$$ab + ac = a(b + c)$$
Where $a$ is the common factor.
Example 1: Numbers Only
Factorise: $6x – 9$
Step 1: Find the HCF.
The biggest number that divides $6$ and $9$ is $[[mathbf{3}]]$.
Step 2: Divide and fill the bracket.
Write the 3 outside:
$3( \dots )$
Divide the first term: $6x [[div]] 3 = 2x$
Divide the second term: $-9 [[div]] 3 = -3$
Final Answer:
$$3(2x – 3)$$
Example 2: Letters Involved
Factorise: $x^2 + 5x$
Step 1: Find the HCF.
There are no common numbers (other than 1).
Both terms contain $x$. So, the factor is $[[mathbf{x}]]$.
Step 2: Divide and fill the bracket.
Write $x$ outside:
$x( \dots )$
Divide the first term: $x^2 [[div]] x = x$
Divide the second term: $5x [[div]] x = 5$
Final Answer:
$$x(x + 5)$$
3. Tricky Terms (Numbers & Letters) 🔢
Sometimes you can pull out both a number and a letter. This is where students often miss marks by not taking out the highest factor.
Factorise: $8x^2 – 12x$
Step 1: Check Numbers.
Factors of 8 and 12: 1, 2, 4. The biggest is $[[mathbf{4}]]$.
Step 2: Check Letters.
Both terms have an $x$. The common letter is $[[mathbf{x}]]$.
Step 3: Combine them.
The total factor is $4x$.
Step 4: Fill the bracket.
$8x^2 [[div]] 4x = 2x$
$-12x [[div]] 4x = -3$
Final Answer:
$$4x(2x – 3)$$
Quick Check: Expand your answer mentally. Does $4x [[times]] 2x$ give you $8x^2$? Does $4x [[times]] -3$ give you $-12x$? If yes, you are correct!
Practice Problems
Try these practice questions to test your speed.
1. Factorise completely:
$$5y + 20$$
Check Answer
Step 1: Find the HCF.
The biggest number that goes into 5 and 20 is $[[mathbf{5}]]$.
Step 2: Divide terms.
$5y [[div]] 5 = y$
$20 [[div]] 5 = 4$
Final Answer:
$$5(y + 4)$$
2. Factorise completely:
$$y^2 - 7y$$
Check Answer
Step 1: Find the HCF.
There are no common numbers. The common letter is $[[mathbf{y}]]$.
Step 2: Divide terms.
$y^2 [[div]] y = y$
$-7y [[div]] y = -7$
Final Answer:
$$y(y - 7)$$
3. Factorise completely:
$$6x^2 + 9x$$
Check Answer
Step 1: Find the HCF.
Numbers: HCF of 6 and 9 is $[[mathbf{3}]]$.
Letters: Common letter is $[[mathbf{x}]]$.
Factor: $3x$.
Step 2: Divide terms.
$6x^2 [[div]] 3x = 2x$
$9x [[div]] 3x = 3$
Final Answer:
$$3x(2x + 3)$$
Interactive Quiz
Can you spot the correct factorisation?