Factors of Polynomials

Introduction to Polynomials

A polynomial is an algebraic expression made using variables, constants and powers.

Examples

• x² + 3x − 4

• 2x³ − 5x² + x + 7

• 4x⁴ − 9

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Degree of a Polynomial

The degree is the highest power of the variable.

Polynomial	Degree
x + 2	1
x² − 5	2
x³ + x	3
2x⁵ − x² + 1	5

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Types of Polynomials

Type	Example
Linear	2x + 3
Quadratic	x² − 4x + 5
Cubic	x³ − 2x + 1
Quartic	x⁴ + 3x² − 7

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Factorisation

Factorisation means writing a polynomial as a product of simpler expressions.

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Common Factor Method

Take out the greatest common factor.

Example

6x² + 9x

= 3x(2x + 3)

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Factorising Quadratics

Basic Form

x² + bx + c

Find two numbers that:

• multiply to give c

• add to give b

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Example

x² + 5x + 6

= (x + 2)(x + 3)

Because:

• 2 × 3 = 6

• 2 + 3 = 5

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Factorising Quadratics with Leading Coefficient

Example

2x² + 7x + 3

Multiply first and last coefficients:

2 × 3 = 6

Find two numbers:

• multiply to give 6

• add to give 7

Numbers:

6 and 1

Rewrite:

2x² + 6x + x + 3

Factorise:

= 2x(x + 3) + 1(x + 3)

= (2x + 1)(x + 3)

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Written and Compiled By Sir Hunain Zia (AYLOTI), World Record Holder With 154 Total A Grades, 11 World Records and 7 Distinctions, Educate A Change.

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Difference of Squares

Formula

a² − b² = (a − b)(a + b)

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Example

x² − 25

= (x − 5)(x + 5)

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Example

9x² − 16

= (3x − 4)(3x + 4)

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Perfect Square Trinomials

Formula 1

a² + 2ab + b² = (a + b)²

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Formula 2

a² − 2ab + b² = (a − b)²

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Example

x² + 6x + 9

= (x + 3)²

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Example

x² − 10x + 25

= (x − 5)²

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Cubic Factorisation

Common Method

1. Find one factor

2. Divide polynomial

3. Factorise remaining quadratic

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Factor Theorem

If:

f(a) = 0

Then:

(x − a) is a factor of f(x)

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Example

Given:

f(x) = x³ − 4x² − x + 4

Check x = 1:

f(1)

= 1 − 4 − 1 + 4

= 0

Therefore:

(x − 1) is a factor.

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Remainder Theorem

If f(x) is divided by (x − a), then remainder = f(a)

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Example

Find remainder when:

2x³ + 3x² − 5

is divided by:

(x − 2)

Solution

f(2)

= 2(2³) + 3(2²) − 5

= 16 + 12 − 5

= 23

Remainder = 23

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Long Division of Polynomials

Used when factorisation is not obvious.

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Example

Divide:

x³ − 2x² − 5x + 6

by:

x − 3

Result:

x² + x − 2

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Synthetic Division

A faster method for dividing polynomials.

Mostly used for cubics and quartics.

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Solving Polynomial Equations

Example

x² − 9 = 0

Factorise:

(x − 3)(x + 3) = 0

Solutions:

x = 3 or x = −3

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Example

x³ − 4x² − x + 4 = 0

Factorise:

(x − 1)(x² − 3x − 4)

Further factorise:

= (x − 1)(x − 4)(x + 1)

Solutions:

• x = 1

• x = 4

• x = −1

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Written and Compiled By Sir Hunain Zia (AYLOTI), World Record Holder With 154 Total A Grades, 11 World Records and 7 Distinctions, Educate A Change.

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Polynomial Identities

Identity 1

(a + b)² = a² + 2ab + b²

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Identity 2

(a − b)² = a² − 2ab + b²

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Identity 3

(a + b)(a − b) = a² − b²

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Identity 4

(a + b)³ = a³ + 3a²b + 3ab² + b³

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Identity 5

(a − b)³ = a³ − 3a²b + 3ab² − b³

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Algebraic Fractions and Factors

Example

x² − 9/x² − x − 6

Factorise numerator:

(x − 3)(x + 3)

Factorise denominator:

(x − 3)(x + 2)

Cancel common factor:

= (x + 3)/(x + 2)

Where:

x ≠ 3, −2

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Conditions for Repeated Factors

Repeated roots occur when the same factor appears multiple times.

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Example

(x − 2)²

Repeated root:

x = 2

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Graph Connections

Linear Factor

Each factor:

(x − a)

gives root:

x = a

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Example

(x − 2)(x + 3)

Roots:

• x = 2

• x = −3

These are x-intercepts on the graph.

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Polynomial Graphs

Quadratic

• Maximum 2 roots

• Parabola

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Cubic

• Maximum 3 roots

• S-shaped graph

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Quartic

• Maximum 4 roots

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Using Graphs to Estimate Factors

If graph crosses x-axis at:

x = 2

then:

(x − 2) is a factor.

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Algebraic Proof Using Factors

Example

Prove:

x³ − x is divisible by 6

Factorise:

x³ − x

= x(x² − 1)

= x(x − 1)(x + 1)

Three consecutive integers always contain:

• one multiple of 2

• one multiple of 3

Therefore expression divisible by 6.

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Written and Compiled By Sir Hunain Zia (AYLOTI), World Record Holder With 154 Total A Grades, 11 World Records and 7 Distinctions, Educate A Change.

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Common Factorisation Mistakes

Mistake	Incorrect	Correct
Wrong signs	x² − 9 = (x − 3)²	(x − 3)(x + 3)
Forgetting common factor	4x² + 8x = (x + 2)(x + 4)	4x(x + 2)
Difference of squares error	a² − b² = (a − b)²	(a − b)(a + b)
Perfect square error	x² + 4 = (x + 2)²	Cannot factorise normally
Wrong factor theorem use	f(a) ≠ 0 but factor assumed	Must equal 0

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Common Expansion Mistakes

Mistake	Incorrect	Correct
Squaring brackets	(a + b)² = a² + b²	a² + 2ab + b²
Missing middle term	(x + 3)(x + 2) = x² + 6	x² + 5x + 6
Cubic expansion error	(a + b)³ incomplete	Use full identity

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Exam Tips

• Always check for common factors FIRST.

• Watch signs carefully.

• Verify factors by expanding again.

• Use factor theorem for cubics.

• Write all roots clearly.

• Do not cancel factors unless fully factorised.

• State restrictions in algebraic fractions.

• Use graphs to estimate roots quickly.

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Written and Compiled By Sir Hunain Zia (AYLOTI), World Record Holder With 154 Total A Grades, 11 World Records and 7 Distinctions, Educate A Change.