Procedure

HCF and LCM of polynomials from their complete factorisations

Algebra • Polynomial • HCF and LCM

Statement

Factor every polynomial completely into irreducible factors. Their HCF is the product of each COMMON irreducible factor raised to the LOWEST power in which it appears; their LCM is the product of each DISTINCT irreducible factor raised to the HIGHEST power in which it appears.

How to use it

1Factor each expression right down to irreducibles first - a quadratic with negative discriminant, such as $x^{2} - 2 x + 4$ , is already irreducible and stops there.
2Do not drop a monomial factor: $x$ pulled out of $x^{3} + 4 x^{2} + 4 x$ is a factor like any other and must appear in the LCM.
3Check the result by confirming that every original expression divides the LCM.

Worked example

Find the LCM of

x^{2} - 1

and

x^{2} + 2 x + 1

Step-by-step

1
$x^{2} - 1 = (x - 1) (x + 1)$
Why: Difference of two squares.
2
$x^{2} + 2 x + 1 = (x + 1)^{2}$
3
$LCM = (x - 1) (x + 1)^{2}$
Why: $(x + 1)$ appears to the power $2$ at most, and $(x - 1)$ to the power $1$ .

Answer

(x - 1) (x + 1)^{2}

Watch out

The rule needs COMPLETE factorisation: leaving

x^{2} + 5 x + 6

unfactored hides the shared

(x + 2)

and inflates the LCM. Taking the highest power where the lowest is wanted (or the reverse) swaps the HCF and the LCM.

Builds on

Factorise a quadratic by splitting the middle term

Also known as

LCM of polynomials, HCF of algebraic expressions, highest power of each factor, lcm hcf of polynomials, gcd of polynomials

Appears in past papers

2018 (II)

2017 (II)

Q8 Q20

2017 (I)

Q41

Statement

Worked example

Appears in past papers

2020 (II)

2020 (I)

2019 (II)

2018 (II)

2017 (II)

2017 (I)