C. Polynomial and Rational Functions · Polynomial Functions and Equations

More Factoring: Sum and Difference of Cubes, Grouping and Quadratic Form

C3.2About 70 minutes

Learning goals

I can factor a sum or difference of cubes and explain where the pattern comes from.
I can factor polynomials by grouping.
I can factor quartics that are quadratic in form, such as x⁴ − 13x² + 36.
I can choose an efficient strategy and factor polynomials of degree up to 4 fully.

Words to know

difference of cubes
An expression a³ − b³, which factors as (a − b)(a² + ab + b²).

sum of cubes
An expression a³ + b³, which factors as (a + b)(a² − ab + b²).

factoring by grouping
Splitting the terms into groups that share a common factor, then factoring out a common binomial.

quadratic in form
A polynomial like ax⁴ + bx² + c that is a quadratic in the variable x².

Shortcuts that save a lot of testing

The factor theorem always works, but some polynomials have patterns that factor them in one step. Remember from Grade 10: a² − b² = (a − b)(a + b) and a² ± 2ab + b² = (a ± b)². Today you add three more tools.

Sum and difference of cubes:

a³ − b³ = (a − b)(a² + ab + b²)
a³ + b³ = (a + b)(a² − ab + b²)

Where do they come from? By the factor theorem, x − b is a factor of x³ − b³ (put x = b). Dividing gives the trinomial x² + bx + b². You can check by expanding: (a − b)(a² + ab + b²) = a³ + a²b + ab² − a²b − ab² − b³ = a³ − b³ ✓.

Why a³ − b³ = (a − b)(a² + ab + b²): take a small cube (side b) out of the corner of a big cube (side a). What is left splits into three boxes, each of depth a − b, with front faces a², ab and b². (Drawn with a = 3, b = 1.)

The SOAP pattern for the signs: the binomial has the same sign as the original, the middle term of the trinomial has the opposite sign, and the last term is always positive.

Worked example

Example 1: Difference of cubes

Factor 8x³ − 27.

Write each term as a cube: 8x³ = (2x)³ and 27 = 3³. So a = 2x and b = 3.
a³ − b³ = (a − b)(a² + ab + b²) = (2x − 3)((2x)² + (2x)(3) + 3²).

8x³ − 27 = (2x − 3)(4x² + 6x + 9)

Worked example

Example 2: Sum of cubes

Factor 125x³ + 64.

125x³ = (5x)³ and 64 = 4³: a = 5x, b = 4.
a³ + b³ = (a + b)(a² − ab + b²) = (5x + 4)(25x² − 20x + 16).
The trinomial does not factor further: its discriminant is (−20)² − 4(25)(16) = 400 − 1600 < 0.

125x³ + 64 = (5x + 4)(25x² − 20x + 16)

Worked example

Example 3: Grouping

Factor x³ + 2x² − 9x − 18.

Group in pairs: (x³ + 2x²) + (−9x − 18).
Common factor in each group: x²(x + 2) − 9(x + 2).
Common binomial: (x + 2)(x² − 9) = (x + 2)(x − 3)(x + 3).

x³ + 2x² − 9x − 18 = (x + 2)(x − 3)(x + 3)

Worked example

Example 4: Quadratic in form

Factor (a) x⁴ − 13x² + 36 (b) x⁴ + 3x² − 4.

(a) Let m = x². Then m² − 13m + 36 = (m − 4)(m − 9). Put x² back: (x² − 4)(x² − 9).
Each is a difference of squares: (x − 2)(x + 2)(x − 3)(x + 3).
(b) m² + 3m − 4 = (m + 4)(m − 1), so (x² + 4)(x² − 1) = (x² + 4)(x − 1)(x + 1). The factor x² + 4 is a sum of squares and does not factor over the real numbers.

(a) (x − 2)(x + 2)(x − 3)(x + 3) (b) (x² + 4)(x − 1)(x + 1)

Worked example

Example 5: Common factor first

Factor 2x⁴ − 16x fully.

Common factor 2x: 2x(x³ − 8).
Difference of cubes: x³ − 8 = (x − 2)(x² + 2x + 4).
x² + 2x + 4 has discriminant 4 − 16 < 0, so it is done.

2x⁴ − 16x = 2x(x − 2)(x² + 2x + 4)

Look for ...	Strategy	Example
a common factor in every term	take it out first, always	2x⁴ − 16x = 2x(x³ − 8)
two terms, both cubes	sum or difference of cubes	x³ + 27 = (x + 3)(x² − 3x + 9)
two terms, both squares, minus	difference of squares	x⁴ − 81 = (x² − 9)(x² + 9)
four terms	try grouping in pairs	x³ − 3x² − 4x + 12
only x⁴, x² and constant terms	substitute m = x² (quadratic form)	x⁴ − 5x² + 4
none of the above	factor theorem (last lesson)	x³ − 2x² − 5x + 6

Choosing a factoring strategy. After each step, check whether any factor can be factored again.

Just for fun

Many students remember the signs in the sum and difference of cubes with the word SOAP, which may be why these are the cleanest factorizations in the unit.

Hands-on activity

Build a³ − b³ with cubes

You need: 27 small cubes of the same size (sugar cubes, dice or building bricks).

Build a 3 by 3 by 3 cube (a = 3). How many small cubes? (a³)
Remove one corner cube (b = 1). How many are left? (a³ − b³)
Split what is left into three boxes, each 2 cubes deep (a − b = 2): one with a 3 by 3 face, one with a 3 by 1 face, and one with a 1 by 1 face. Count the cubes in each box.
Check that the total matches (a − b)(a² + ab + b²) with a = 3 and b = 1.

What to look for: 27 − 1 = 26 cubes. The boxes hold 3 × 3 × 2 = 18, 3 × 1 × 2 = 6 and 1 × 1 × 2 = 2 cubes, and 18 + 6 + 2 = 26. The formula gives (3 − 1)(9 + 3 + 1) = 2 × 13 = 26 ✓.

Your turn

Factor x³ − 64.

Factor x³ + 1.

Factor 27x³ + 8.

Factor x³ − 3x² − 4x + 12 by grouping.

Factor x⁴ − 5x² + 4.

Factor 3x³ − 24 fully.

Factor x⁴ − 81 fully.

More practice: warm-up

What is the cube root of 8x³?

125 is the cube of what number?

In a³ + b³ = (a + b)(a² − ab + b²), what is the sign of the ab term?

Does x² + 2x + 4 factor over the real numbers?

What is the binomial factor of x³ + 27?

More practice: on your own

Factor 64x³ − 1.

Factor 2x³ + 250 fully.

Factor 3x³ − x² + 6x − 2.

Factor x⁴ − 10x² + 16 as far as possible over the integers.

Factor x⁴ + x³ − 8x − 8 fully.

Challenge

Factor x⁶ − 1 fully over the integers. (Hint: start with a difference of squares.)

Factor (x + 1)³ − 8 and check by expanding.

Explain why the trinomial factor x² + bx + b² of x³ − b³ (b ≠ 0) never factors over the real numbers.

Check your understanding

Quiz

Which is the factored form of x³ − 125?

Which is the factored form of 8x³ + 1?

Which factorization of x³ + 5x² − 4x − 20 is correct?

How does x⁴ − 17x² + 16 factor fully?

Which expression cannot be factored over the real numbers?

What should you do first when factoring 5x⁴ − 40x?

For parents and teachers

Big idea (C3.2): besides the factor theorem, polynomials can often be factored with patterns: sum and difference of cubes, grouping, and treating x⁴ as (x²)². Choosing the right strategy saves a lot of time.

How to help: Ask your teen to check each factorization by expanding it, or by substituting a number such as x = 2 into both forms. For the cubes patterns, the word SOAP helps them remember the signs.

Common difficulties:

Writing (a − b)(a² + 2ab + b²) or other incorrect trinomials for cubes.
Trying to factor x² + 4 (a sum of squares does not factor over the real numbers).
Forgetting the common factor at the start.
Stopping early, for example leaving x² − 9 unfactored.