This cheat sheet covers three special factoring patterns that appear often in Algebra 1 and Algebra 2: difference of squares, sum of cubes, and difference of cubes. Students need these patterns because they make many polynomial expressions faster to factor and simplify. Recognizing the structure of an expression helps avoid long trial-and-error factoring.
These identities are also useful when solving equations, simplifying rational expressions, and preparing for higher algebra.
Key Facts
- The difference of squares pattern is .
- A difference of squares must have two perfect square terms separated by subtraction, such as .
- The sum of cubes pattern is .
- The difference of cubes pattern is .
- For cubes, the binomial factor uses the same sign as the original expression: gives and gives .
- For cubes, the trinomial signs follow the pattern SOAP: Same sign, Opposite sign, Always Positive.
- Always check for a greatest common factor first, such as .
- You can verify any factoring identity by multiplying the factors back together to get the original expression.
Vocabulary
- Difference of Squares
- A factoring pattern for subtracting two perfect squares, written as .
- Sum of Cubes
- A factoring pattern for adding two perfect cubes, written as .
- Difference of Cubes
- A factoring pattern for subtracting two perfect cubes, written as .
- Perfect Square
- A number or expression that can be written as something squared, such as or .
- Perfect Cube
- A number or expression that can be written as something cubed, such as or .
- Greatest Common Factor
- The largest factor shared by all terms in an expression, often factored out before using special patterns.
Common Mistakes to Avoid
- Factoring as is wrong because the difference of squares pattern only works for subtraction, not addition.
- Using the wrong signs in a cube formula is wrong because factors as , while factors as .
- Forgetting to find the greatest common factor first can leave the expression only partly factored, such as writing without first factoring .
- Mistaking a non-perfect cube for a cube pattern is wrong because terms like are not perfect cubes unless every factor fits a cube form.
- Dropping variables or exponents during substitution is wrong because equals , not .
Practice Questions
- 1 Factor completely: .
- 2 Factor completely: .
- 3 Factor completely: .
- 4 Explain why cannot be factored using the difference of squares pattern over the real numbers.
Understanding Difference of Squares and Sum and Difference of Cubes
These patterns work because multiplication creates middle terms that either cancel or combine in predictable ways. When two conjugate binomials are multiplied, one has addition and the other has subtraction. The positive cross product and negative cross product cancel.
Only the square of the first term and the negative square of the second term remain. This explains why a difference of squares has no middle term.
It also explains why an expression with addition between two squares does not use this pattern over the real numbers. For example, a square plus a square does not split into two simple real binomials.
Cube patterns need more care because multiplying a binomial by a trinomial produces several middle terms. The signs in the trinomial are chosen so that the unwanted terms cancel. A useful memory aid is SOAP, but students should know what it is protecting them from.
The last term of the trinomial stays positive in both cube patterns. Changing it to negative creates the wrong final product.
The middle term changes sign depending on whether the original cubes were added or subtracted. Many errors happen when a student remembers the first factor but guesses the signs in the second factor.
Before using any special pattern, inspect the whole expression. Take out every common numerical factor and variable factor first. This can reveal a pattern that was hidden at the start.
Then check whether each remaining term is truly a perfect square or perfect cube. A term such as eight times x cubed is a perfect cube because it is two cubed times x cubed. A term such as twelve times x squared is not a perfect square term, even though x squared is a square.
Powers provide another clue. Even exponents often point toward squares, while exponents that are multiples of three can point toward cubes. Coefficients must fit the same pattern.
These factorizations appear when solving polynomial equations. After moving all terms to one side, factoring can turn one difficult equation into simpler factors set equal to zero. They appear in rational expressions too, where a shared factor may cancel only after the numerator or denominator has been factored completely.
Always state restrictions before cancelling, since a value that makes an original denominator zero remains excluded. The safest final check is multiplication.
Multiply the first factor through the second factor, combine like terms, and compare every term with the starting expression. This check catches sign mistakes quickly and builds confidence that the factorization is complete.