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Difference of Squares and Sum and Difference of Cubes cheat sheet - grade 9-10

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Math Grade 9-10

Difference of Squares and Sum and Difference of Cubes Cheat Sheet

A printable reference covering difference of squares, sum of cubes, difference of cubes, factoring patterns, and sign rules for grades 9-10.

TopicsAlgebraPatternsExponentsOrder of OperationsMathematicsStudy Guide
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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 a2−b2=(a−b)(a+b)a^2 - b^2 = (a - b)(a + b).
  • A difference of squares must have two perfect square terms separated by subtraction, such as x2−25x^2 - 25.
  • The sum of cubes pattern is a3+b3=(a+b)(a2−ab+b2)a^3 + b^3 = (a + b)(a^2 - ab + b^2).
  • The difference of cubes pattern is a3−b3=(a−b)(a2+ab+b2)a^3 - b^3 = (a - b)(a^2 + ab + b^2).
  • For cubes, the binomial factor uses the same sign as the original expression: a3+b3a^3 + b^3 gives (a+b)(a + b) and a3−b3a^3 - b^3 gives (a−b)(a - b).
  • 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 2x3−16=2(x3−8)2x^3 - 16 = 2(x^3 - 8).
  • 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 a2−b2=(a−b)(a+b)a^2 - b^2 = (a - b)(a + b).
Sum of Cubes
A factoring pattern for adding two perfect cubes, written as a3+b3=(a+b)(a2−ab+b2)a^3 + b^3 = (a + b)(a^2 - ab + b^2).
Difference of Cubes
A factoring pattern for subtracting two perfect cubes, written as a3−b3=(a−b)(a2+ab+b2)a^3 - b^3 = (a - b)(a^2 + ab + b^2).
Perfect Square
A number or expression that can be written as something squared, such as 49=7249 = 7^2 or x2=x2x^2 = x^2.
Perfect Cube
A number or expression that can be written as something cubed, such as 27=3327 = 3^3 or 8x3=(2x)38x^3 = (2x)^3.
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 a2+b2a^2 + b^2 as (a+b)(a−b)(a + b)(a - b) 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 a3+b3a^3 + b^3 factors as (a+b)(a2−ab+b2)(a + b)(a^2 - ab + b^2), while a3−b3a^3 - b^3 factors as (a−b)(a2+ab+b2)(a - b)(a^2 + ab + b^2).
  • Forgetting to find the greatest common factor first can leave the expression only partly factored, such as writing 2x3−162x^3 - 16 without first factoring 2(x3−8)2(x^3 - 8).
  • Mistaking a non-perfect cube for a cube pattern is wrong because terms like 12x312x^3 are not perfect cubes unless every factor fits a cube form.
  • Dropping variables or exponents during substitution is wrong because (2x)3(2x)^3 equals 8x38x^3, not 2x32x^3.

Practice Questions

  1. 1 Factor completely: x2−81x^2 - 81.
  2. 2 Factor completely: 27x3+6427x^3 + 64.
  3. 3 Factor completely: 2x3−542x^3 - 54.
  4. 4 Explain why x2+16x^2 + 16 cannot be factored using the difference of squares pattern over the real numbers.