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Bucks County Community College Tutoring CenterFactor the Sum and Difference of Two Cubes1.Formulas for factoring the Sumand Differenceof two cubes:Sum: a³+b³=(a+b)(a²-ab+b²)Difference: a³-b³=(a-b)(a²+ab+b²)Note: Keep in mind that the middle of the trinomial is always opposite the sign of the binomial2.Identification of Sum and Difference in the given problem:a³+b³ or a³-b³↓ ↓Ex: x³+8 27x³-8↓ ↓x³+2³ (3x)³-2³↓ ↓ ↓ ↓let:a=x b=2a=3x b=2(Thecubed roots of each term in the original)Sample of perfect cubes:1x327x38x3y38x327x664x3y364x9125x6y3125The exponents must be divisible by 3 for a perfect cube3.Match it to the sum or difference formulas:Use your “a” and “b” values to match “a” and “b” in the formula you have chosen: Factor: x3+ 8Sum: a³+b³ = (a+b) (a²-ab+b²)↑ ↑ ↑ ↑↑ ↑ ↑(cube roots x 2)(x+2) (x²-2x+2²)So: x3+8 =x3+23= (x+2) (x²-2x+4)Note:the middle sign of the trinomial is opposite of the binomial3.To prove your answer is right multiply (x+2)(x²-2x+4) → using the distributive property: ↔(x+2)(x²-2x+4)↔So: x³-2x²+4x+2x²-4x+8 Simplify by canceling like terms You get x³+8whichprovesthat your answer is correct.
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