CDS Mathematics · Polynomials

HCF and LCM of Polynomials

Factorise every polynomial; the HCF is the product of the common factors to their lowest power, the LCM of all factors to their highest power, and HCF × LCM equals the product of two polynomials.

All Polynomials notes CDS Mathematics strategy

Why this matters

Twenty-five PYQs, the largest page in the chapter and one of the most reliable in the paper. Everything is factorising: once each polynomial is in factors, the HCF and LCM are read off, and the product rule finds a missing polynomial.

Concept 1 of 3: Reading off the HCF and LCM

The HCF of numbers uses the common primes to their smallest powers; the LCM uses every prime to its largest power. Polynomials work the same way with irreducible factors in place of primes.

Definition

Factorise each polynomial completely.
HCF = product of factors common to ALL, each to the LOWEST power present.
LCM = product of EVERY factor appearing, each to the HIGHEST power present.
Useful factorisations: $x^6 - y^6 = (x - y)(x + y)(x^2 + xy + y^2)(x^2 - xy + y^2)$ , $x^8 + x^4 + 1 = (x^4 + x^2 + 1)(x^4 - x^2 + 1)$ .

HCF and LCM

\text{HCF}: \text{common factors, lowest power}; \quad \text{LCM}: \text{all factors, highest power}

Worked example

Find the HCF and LCM of

x^2 - 1

x^2 + 2x + 1

and

x^3 + 1

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2021 · CDS (I) 2021 — Elementary Mathematics · Q16Moderate

Example 1 · Polynomials · HCF and LCM of Polynomials

What is the HCF of

x^3 - 19x + 30

and

x^2 - 5x + 6

Lowest power for the HCF

(x + 1)^3

and

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

(x + 1)^2

, not

(x + 1)^3

. Taking the higher power belongs to the LCM.

Drill 14 more on reading off the hcf and lcm

Concept 2 of 3: HCF × LCM = product

Every factor of two polynomials appears once in the HCF-and-LCM pair for each time it appears in the two polynomials. So their product equals HCF × LCM, which finds a missing polynomial by one division.

Definition

For two polynomials: $p(x)\,q(x) = \text{HCF} \times \text{LCM}$ .
So $q(x) = \dfrac{\text{HCF} \times \text{LCM}}{p(x)}$ .
The HCF must divide the LCM; if it does not, the data are inconsistent.
If the HCF is $1$ , the LCM is the product.

Product rule

p(x)\,q(x) = \text{HCF}(p, q)\times\text{LCM}(p, q)

Worked example

The HCF of two polynomials is

x - 1

and their LCM is

(x - 1)(x + 2)(x - 3)

. One is

x^2 + x - 2

. Find the other.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2018 · CDS (II) 2018 — Elementary Mathematics · Q8Moderate

Example 2 · Polynomials · HCF and LCM of Polynomials

HCF and LCM of two polynomials are (x + 3) and

(x^3 - 9x^2 - x + 105)

respectively. If one of the two polynomials is

x^2 - 4x - 21

, then the other is

Only for two polynomials

p q = \text{HCF}\times\text{LCM}

is true for a pair. For three polynomials the product is generally not HCF

\times

LCM.

Drill 5 more on hcf × lcm = product

Concept 3 of 3: Unknowns from a given HCF

x - k

is a common factor, then

k

is a root of both polynomials. That gives two equations; subtracting them removes the

x^2

term.

Definition

$x - k$ divides both $\Rightarrow$ substitute $x = k$ in each: two equations.
Subtracting two monic quadratics' equations at $x = k$ gives $k$ directly: $(a - c)k + (b - d) = 0$ .
A given quadratic HCF: factorise it and use both roots.

Common root

k^2 + ak + b = 0,\ k^2 + ck + d = 0 \;\Rightarrow\; k = \dfrac{d - b}{a - c}

Worked example

x - 2

is the HCF of

x^2 + ax - 6

and

x^2 - 5x + b

. Find

a

and

b

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2026 · CDS (I) 2026 — Elementary Mathematics · Q6Moderate

Example 3 · Polynomials · HCF and LCM of Polynomials

(x - 5)

is the HCF of

x^2 - x - p

and

x^2 - qx - 10

, then what is the value of

(p + q)

x + k has root −k

If the HCF is

x + k

, substitute

x = -k

. The answer for

k

then comes out with the opposite sign to the root.

Drill 3 more on unknowns from a given hcf

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (3)

Reading off the HCF and LCM
HCF and LCM
$\text{HCF}: \text{common factors, lowest power}; \quad \text{LCM}: \text{all factors, highest power}$
HCF × LCM = product
Product rule
$p(x)\,q(x) = \text{HCF}(p, q)\times\text{LCM}(p, q)$
Unknowns from a given HCF
Common root
$k^2 + ak + b = 0,\ k^2 + ck + d = 0 \;\Rightarrow\; k = \dfrac{d - b}{a - c}$

Watch out for (3)

Lowest power for the HCF
→ Reading off the HCF and LCM
Only for two polynomials
→ HCF × LCM = product
x + k has root −k
→ Unknowns from a given HCF

Test yourself on Polynomials

20 past CDS questions from this chapter, timed at 24 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

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