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Algebra 223Updated Sep 8, 20262 pages

Understanding the Sum and Difference of Cubes

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Factoring polynomial expressions is like solving math puzzles that reveal hidden patterns. This guide will help you master factoring the sum and difference of cubes, along with related factoring techniques that make algebra problems much easier to solve.
Sum and Difference of Cubes – page 1

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Sum and Difference of Cubes

Ever wondered how to break down complex cubic expressions? The key is memorizing two powerful formulas. For the difference of cubes, use: a3−b3=(a−b)(a2+ab+b2)a^3 - b^3 = (a - b)(a^2 + ab + b^2)

For the sum of cubes, the pattern is slightly different: a3+b3=(a+b)(a2−ab+b2)a^3 + b^3 = (a + b)(a^2 - ab + b^2)

Let's see this in action! When factoring x3−27x^3 - 27, identify it as x3−33x^3 - 3^3. Using the difference of cubes formula, this becomes (x−3)(x2+3x+9)(x-3)(x^2+3x+9). The first factor is simple, while the second never factors further.

💡 Quick Tip: When factoring expressions like 64a6−b664a^6 - b^6, you can either factor as a difference of cubes first or as a difference of squares. Both approaches work, but choosing wisely can save you time!

Remember that sometimes you'll need to factor out common terms first. For example, with 2a4+16a2a^4 + 16a, first factor out 2a2a to get 2a(a3+8)2a(a^3+8). Then the expression inside parentheses is a sum of cubes that factors as 2a(a+2)(a2−2a+4)2a(a+2)(a^2-2a+4).

Sum and Difference of Cubes – page 2

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Factoring by Grouping Terms

Factoring by grouping is like solving a puzzle by rearranging pieces. This technique is super useful when expressions don't immediately look like standard patterns.

Start by looking for ways to rewrite the expression as a difference of squares (a+b)(a−b)(a+b)(a-b) or as a perfect square minus another term. For example, with x2+6x+9−y2x^2+6x+9-y^2, recognize that the first three terms form a perfect square trinomial: (x+3)2−y2(x+3)^2-y^2. This becomes (x+3+y)(x+3−y)(x+3+y)(x+3-y).

Sometimes you'll need to rearrange terms to spot the pattern. For expression 81−x2+2xy−y281-x^2+2xy-y^2, group the last three terms: 81−(x2−2xy+y2)81-(x^2-2xy+y^2), which simplifies to 81−(x−y)281-(x-y)^2. This is a difference of squares that factors as (9+x−y)(9−x+y)(9+x-y)(9-x+y).

🔍 Strategy Alert: When factoring expressions with four terms, try grouping them into pairs. For example, in 7a2x−6a2−7x+67a^2x-6a^2-7x+6, group terms with common factors to get a2(7x−6)−1(7x−6)a^2(7x-6)-1(7x-6), which equals (a2−1)(7x−6)(a^2-1)(7x-6).

For trickier expressions like 49y2−x2−10x−2549y^2-x^2-10x-25, recognize that the last three terms form (x+5)2(x+5)^2 with a negative sign. Rearranging gives 49y2−(x+5)249y^2-(x+5)^2, which factors as (7y+x+5)(7y−x−5)(7y+x+5)(7y-x-5).

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