Factor $3x^4 - 24x$ fully.

$3x^4 - 24x = 3x(x - 2)(x^2 + 2x + 4)$

Factor $x^6 - 64$ fully.

$x^6 - 64 = (x - 2)(x + 2)(x^2 + 2x + 4)(x^2 - 2x + 4)$

Factor $(x + 2)^3 - 27x^3$ fully.

$(x + 2)^3 - 27x^3 = 2(1 - x)(13x^2 + 10x + 4)$

Grade 12 · MHF4U · Advanced Functions

MHF4U: Sum and Difference of Cubes

A sum or difference of cubes factors every time, by a pattern worth memorising: $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$ and $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$ . Everything else is recognising $a$ and $b$ .

How It Works

The two patterns differ only in signs, and a mnemonic keeps them straight: SOAP, for Same, Opposite, Always Positive. The binomial takes the same sign as the original, the middle term of the trinomial takes the opposite sign, and the last term is always positive. Expand $(a + b)(a^2 - ab + b^2)$ once to see the middle terms cancel.

To use the pattern you need both terms as cubes. Know the perfect cubes to $1000$ : $1, 8, 27, 64, 125, 216, 343, 512, 729, 1000$ . For variables, the exponent must be a multiple of $3$ : $x^6 = (x^2)^3$ and $8x^3 = (2x)^3$ . Write $a$ and $b$ down explicitly before substituting; most sign errors come from skipping that.

Always take out a common factor first. $3x^4 - 24x$ does not look like a difference of cubes until you pull out $3x$ and see $x^3 - 8$ . And when an expression is both a difference of squares and a difference of cubes, like $x^6 - 64$ , do squares first: it leads to a complete factoring, while cubes first leaves $x^4 + 4x^2 + 16$ , which does factor but not by any pattern you know.

The trinomial $a^2 \pm ab + b^2$ does not factor over the real numbers. Its discriminant is $b^2 - 4b^2 = -3b^2$ , which is negative. So once you have it, stop. On a test, factor fully ends there.

The same patterns solve equations: $x^3 - 8 = 0$ becomes $(x - 2)(x^2 + 2x + 4) = 0$ , and the only real root is $x = 2$ , because the quadratic factor has no real zeros.

5 Worked Examples

Easy first, test level last. Try each one on paper, then open the solution.

Easy
Factor $x^3 + 27$ .
Show solution
1. $27 = 3^3$ , so $a = x$ and $b = 3$ , a sum of cubes.
2. $(a + b)(a^2 - ab + b^2) = (x + 3)(x^2 - 3x + 9)$ .
Answer: $x^3 + 27 = (x + 3)(x^2 - 3x + 9)$
Easy
Factor $8x^3 - 125$ .
Show solution
1. $8x^3 = (2x)^3$ and $125 = 5^3$ , so $a = 2x$ and $b = 5$ , a difference of cubes.
2. $(a - b)(a^2 + ab + b^2) = (2x - 5)((2x)^2 + (2x)(5) + 5^2)$ .
Answer: $8x^3 - 125 = (2x - 5)(4x^2 + 10x + 25)$
Medium
Factor $3x^4 - 24x$ fully.
Show solution
1. Common factor first: $3x(x^3 - 8)$ .
2. $x^3 - 8$ is a difference of cubes with $a = x$ , $b = 2$ : $(x - 2)(x^2 + 2x + 4)$ .
3. The trinomial has discriminant $4 - 16 < 0$ , so it does not factor further.
Answer: $3x^4 - 24x = 3x(x - 2)(x^2 + 2x + 4)$
Medium
Factor $x^6 - 64$ fully.
Show solution
1. It is a difference of squares, $(x^3)^2 - 8^2$ , and a difference of cubes, $(x^2)^3 - 4^3$ . Squares first.
2. $(x^3 - 8)(x^3 + 8)$ .
3. Each bracket is a sum or difference of cubes: $x^3 - 8 = (x - 2)(x^2 + 2x + 4)$ and $x^3 + 8 = (x + 2)(x^2 - 2x + 4)$ .
Answer: $x^6 - 64 = (x - 2)(x + 2)(x^2 + 2x + 4)(x^2 - 2x + 4)$
Test level
Factor $(x + 2)^3 - 27x^3$ fully.
Show solution
1. A difference of cubes with $a = x + 2$ and $b = 3x$ .
2. Binomial: $a - b = (x + 2) - 3x = 2 - 2x$ .
3. Trinomial: $a^2 + ab + b^2 = (x + 2)^2 + 3x(x + 2) + 9x^2 = x^2 + 4x + 4 + 3x^2 + 6x + 9x^2 = 13x^2 + 10x + 4$ .
4. Take $2$ out of the binomial: $2(1 - x)(13x^2 + 10x + 4)$ . The discriminant of the trinomial is $100 - 208 < 0$ , so it is done.
Answer: $(x + 2)^3 - 27x^3 = 2(1 - x)(13x^2 + 10x + 4)$

Learn It on Martian Lab

Martian Lab teaches this in the Factor Theorem and Rational Zeros lesson of its MHF4U Polynomial Equations unit. Watch it, then take the Polynomial Equations practice tests to see if it stuck.

Watch the Factor Theorem and Rational Zeros lesson free

Keep Going

Free Polynomial Equations practice test, with solutions

More Polynomial Equations topics: Long and Synthetic Division · Remainder and Factor Theorem · Solving Polynomial Equations · Polynomial Inequalities

More MHF4U topics: Vertical and Horizontal Asymptotes · Proving Trig Identities · Solving Trig Equations · Laws of Logarithms · Solving Logarithmic Equations · End Behaviour and Degree · Finding the Equation of a Polynomial · Even and Odd Functions · Holes in Rational Functions · Graphing Rational Functions · Rational Equations and Inequalities · Radian Measure · Unit Circle and Special Angles · Compound Angle Formulas · Double Angle Formulas · Graphing Sine and Cosine with Transformations · Graphs of Tan, Sec, Csc and Cot · Sinusoidal Word Problems · Solving Exponential Equations · Graphing Logarithmic Functions · Exponential and Log Word Problems · Sums, Products and Quotients of Functions · Composition of Functions · Average vs Instantaneous Rate of Change

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