Factorising with Negatives
When the first term of an expression is negative, it is usual to take out a negative highest common factor so that the bracket starts with a positive term. In NSW Year 8 maths this means \(-6x + 15\) becomes \(-3(2x - 5)\) (NESA outcome MA4‑ALG‑C‑01).
You will factorise expressions with numbers, pronumerals and powers, such as \(-8xy^2 + 4x^2y\), remembering that dividing by a negative changes the sign of every term in the bracket. You will check each answer by expanding it and correct a student's error. Negative terms turn up whenever money is owed, a tank drains or a temperature falls. It is part of Year 8 Algebra.
Theory
When the first term is negative, take out a negative HCF so the bracket starts with a positive term. The HCF of \(-4x^2y\) and \(2x\) can be \(2x\) or \(-2x\); with \(-2x\): \(-4x^2y + 2x = -2x(2xy - 1)\). Dividing by a negative changes every sign inside. This Year 8 maths page shows how.
A HCF can be positive or negative. \(-4x^2y = -2x \times 2xy\) and \(2x = -2x \times (-1)\), so \(-2x\) is a common factor, just as \(2x\) is.
Three steps. Find the HCF (number and pronumerals, lowest powers) and make it negative. Divide each term by it. Write it outside the bracket with the answers inside.
Dividing by a negative flips every sign: \(-6x \div (-3) = 2x\) and \(15 \div (-3) = -5\), so \(-6x + 15 = -3(2x - 5)\). A term divided by itself gives \(1\): \(-4k - 4kn = -4k(1 + n)\).
The HCF may be just \(-1\): \(-m + 7 = -(m - 7)\). Check every answer by expanding, using \((-) \times (-) = (+)\).
Positive or negative HCF:
| Expression | Positive HCF | Negative HCF |
|---|---|---|
| \(-12x + 8\) | \(4(-3x + 2)\) | \(-4(3x - 2)\) |
| \(-6a - 15\) | \(3(-2a - 5)\) | \(-3(2a + 5)\) |
| \(-8y^2 + 20y\) | \(4y(-2y + 5)\) | \(-4y(2y - 5)\) |
How to factorise with a negative HCF
- Find the HCF (ignore signs), then put a minus in front.
- Divide each term by the negative HCF; every sign flips.
- Write HCF(answers) and check by expanding.
\(-4x^2y = -2x \times 2xy\) and \(2x = -2x \times (-1)\).
| \(-4x^2y + 2x\) | \(=\) | \(-2x(2xy - 1)\) |
\(-2x(2xy - 1)\)
| \(\) | \(=\) | \(-3xyz \times (-y) + (-3xyz) \times 3\) |
| \(\) | \(=\) | \(-3xyz(-y + 3)\) |
| \(\) | \(=\) | \(-3xyz(3 - y)\) |
\(-3xyz(3 - y)\)
(i) \(-5a + 30\)
(ii) \(-14m - 35\)
| \(-5a + 30\) | \(=\) | \(-5(a - 6)\) |
| \(-14m - 35\) | \(=\) | \(-7(2m + 5)\) |
(i) \(-5(a - 6)\)
(ii) \(-7(2m + 5)\)
(i) \(-6y + 10yz\)
(ii) \(-t^2 - t\)
| \(-6y + 10yz\) | \(=\) | \(-2y(3 - 5z)\) |
| \(-t^2 - t\) | \(=\) | \(-t(t + 1)\) |
(i) \(-2y(3 - 5z)\)
(ii) \(-t(t + 1)\)
(i) \(-20a^2b + 15ab^2\)
(ii) \(-24p^2qr + 8pqr\)
| \(-20a^2b + 15ab^2\) | \(=\) | \(-5ab(4a - 3b)\) |
| \(-24p^2qr + 8pqr\) | \(=\) | \(-8pqr(3p - 1)\) |
(i) \(-5ab(4a - 3b)\)
(ii) \(-8pqr(3p - 1)\)
| \(-6g(3h + 4)\) | \(=\) | \(-18gh - 24g\) |
That is not \(-18gh + 24g\). Since \(24g \div (-6g) = -4\), the answer is \(-6g(3h - 4)\).
Common pitfalls
Frequently asked questions
When should you take out a negative HCF?
When the first term is negative, or when the question asks for it.
What happens to the signs inside the bracket?
They all change: \(-10a + 5 = -5(2a - 1)\).
Can the HCF be -1?
Yes: \(-p + 3 = -(p - 3)\).
How do you check a negative factorisation?
Expand it: \(-5(2a - 1) = -10a + 5\).
Is a negative HCF still a common factor?
Yes. Both \(2x\) and \(-2x\) divide \(-4x^2y\) and \(2x\) exactly.