Sum and difference of cubes; factoring by grouping
Sum and Difference of Cubes; Grouping
Sum and Difference of Cubes; Grouping is a topic in Polynomial Functions in the Texas Essential Knowledge and Skills (Algebra II, §111.40). It is aligned to Standard 2A.7(D), which requires students to determine the linear factors of a polynomial of degree three and four using algebraic methods.
The cube patterns \(a^3\pm b^3\) factor with the SOAP sign rule, and four-term polynomials factor by grouping.
Theory
Two cube patterns and grouping extend the factoring toolkit:
The signs follow SOAP: Same, Opposite, Always Positive.
The cube patterns:
How to factor cubes
- Take any GCF first.
- Write each term as a perfect cube \(a^3,\ b^3\).
- Apply the sum or difference pattern with SOAP signs.
- For four terms, factor by grouping instead.
Write as \(x^3+2^3\) and apply the pattern.
| \(x^3+2^3\) | \(=\) | \((x+2)(x^2-2x+4)\) |
Write as \((3x)^3-1^3\).
| \((3x)^3-1^3\) | \(=\) | \((3x-1)(9x^2+3x+1)\) |
Take the GCF \(2\) first.
| \(2x^3+16\) | \(=\) | \(2(x^3+8)\) |
| \(=\) | \(2(x+2)(x^2-2x+4)\) |
Group and factor each pair.
| \(x^2(x-1)+2(x-1)\) | ||
| \(=\) | \((x-1)(x^2+2)\) |
Common pitfalls
Frequently asked questions
How do you factor a sum of cubes?
\(a^3+b^3=(a+b)(a^2-ab+b^2)\).
What is the SOAP rule?
Same, Opposite, Always Positive — the signs of the three parts.
When do you factor by grouping?
When a polynomial has four terms that pair into common factors.
Does the quadratic factor from cubes factor again?
Usually not over the real numbers.