Factorising writes an expression as a product. Take out the highest common factor, then (Extended) factorise quadratics, the difference of two squares, perfect squares, groups of four terms and simple cubics.
Always take out the highest common factor first, then look inside the bracket for a pattern: a quadratic, a difference of two squares or a perfect square. Factorise fully means nothing left can be factorised again. Check by expanding.
Section 01
The highest common factor.
In the lab choose Common factor and press Next step to see the highest common factor of the numbers, then of each letter, come out in front. Press New question and try to say the factor before the lab shows it.
Numbers: take out the HCF of the coefficients, e.g. 6 from \(12\) and \(18\).
Letters: take out each letter to its lowest power that appears in every term, e.g. \(a^{2}\) from \(a^{3}\) and \(a^{2}\).
Inside the bracket: divide each term by the factor you took out.
Check: expanding your answer must give the original expression.
Worked example Two terms
Factorise fully \(9x^{2} + 15xy\).
HCF of 9 and 15 is 3; both terms contain \(x\); only one contains \(y\). So the HCF is \(3x\).
HCF \(= 7ab^{2}\). \(14a^{2}b^{3} \div 7ab^{2} = 2ab\) and \(21ab^{2} \div 7ab^{2} = 3\). So \(7ab^{2}(2ab + 3)\).
Examiner trap
\(3(3x^{2} + 5xy)\) is factorised, but not fully: \(x\) is still common inside the bracket. 'Factorise fully' loses a mark for this.
Section 02
Quadratics (Extended).
To factorise \(x^{2} + bx + c\), find two numbers that multiply to \(c\) and add to \(b\). When the \(x^{2}\) term has a number in front, \(ax^{2} + bx + c\), find two numbers that multiply to \(ac\) and add to \(b\), split the middle term, and group.
Worked example Leading coefficient 1 (Extended)
Factorise \(x^{2} - 2x - 15\).
Need two numbers with product \(-15\) and sum \(-2\).
Worked example Leading coefficient not 1 (Extended)
Factorise \(6x^{2} - x - 2\).
\(ac = 6 \times (-2) = -12\). Two numbers with product \(-12\) and sum \(-1\): \(-4\) and \(3\).
Split the middle term: \(6x^{2} - 4x + 3x - 2\).
Group: \(2x(3x - 2) + 1(3x - 2)\).
The bracket \((3x - 2)\) is common.
Answer: \((3x - 2)(2x + 1)\)
Your turn
(Extended) Factorise \(3x^{2} - 10x + 8\).
\(ac = 24\); numbers with product 24 and sum \(-10\): \(-6\) and \(-4\). \(3x^{2} - 6x - 4x + 8 = 3x(x - 2) - 4(x - 2) = (3x - 4)(x - 2)\).
Examiner trap
Getting the right numbers with the wrong signs: \((x + 5)(x - 3) = x^{2} + 2x - 15\), not \(x^{2} - 2x - 15\). Expand to check the middle term every time.
Stage 1 · Learn
Check your understanding
Four quick checks. Each correct answer earns XP and records this skill on your profile.
Quick check+10 XP
Factorise fully \(6x + 15\).
Quick check+10 XP
Factorise fully \(4x^{2} - 10x\).
Quick check+10 XP
(Extended) Factorise \(x^{2} + 5x + 6\).
Quick check+10 XP
(Extended) Factorise \(x^{2} - 49\).
Section 03
Special patterns (Extended).
Learn to recognise these shapes on sight. Each one is an expansion you already know, read from right to left.
Pattern
Factorised
Example
\(a^{2}x^{2} - b^{2}y^{2}\)
\((ax - by)(ax + by)\)
\(9x^{2} - 25y^{2} = (3x - 5y)(3x + 5y)\)
\(a^{2} + 2ab + b^{2}\)
\((a + b)^{2}\)
\(4x^{2} + 12x + 9 = (2x + 3)^{2}\)
\(ax + bx + kay + kby\)
\((a + b)(x + ky)\)
grouping in pairs
\(ax^{3} + bx^{2} + cx\)
\(x(ax^{2} + bx + c)\)
take out \(x\), then factorise the quadratic
Worked example Grouping (Extended)
Factorise \(6ax + 3bx + 4ay + 2by\).
Group the first two and the last two: \(3x(2a + b) + 2y(2a + b)\).
Both groups share the bracket \((2a + b)\).
Answer: \((2a + b)(3x + 2y)\)
Worked example A cubic (Extended)
Factorise completely \(2x^{3} - 8x^{2} - 10x\).
Common factor \(2x\): \(2x(x^{2} - 4x - 5)\).
Product \(-5\), sum \(-4\): \(-5\) and \(1\).
So \(x^{2} - 4x - 5 = (x - 5)(x + 1)\).
Answer: \(2x(x - 5)(x + 1)\)
Your turn
(Extended) Factorise completely \(2x^{2} - 50\).
Common factor 2: \(2(x^{2} - 25)\). Difference of two squares: \(2(x - 5)(x + 5)\).
Examiner trap
\(x^{2} + 25\) does NOT factorise. The difference of two squares needs a minus sign between two perfect squares.
Stage 2 · Exam
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