Factor sums and differences of cubes

Learning Objectives Identify binomials that are sums or differences of perfect cubes. Recall and correctly state the formulas for factoring a³ + b³ and a³ - b³. Apply the appropriate formula to factor cubic binomials. Factor out a Greatest Common Factor (GCF) before applying the sum or difference of cubes formulas. Factor expressions with higher-order exponents that fit the cubic pattern, such as x⁶ - y⁹. Verify their factorization by expanding the resulting factors. Use factoring to find the real and complex roots of cubic equations. How can you algebraically deconstruct a perfect 3D cube into its fundamental factors? 🧊 Let's find out! This tutorial focuses on two special factoring patterns: the sum of cubes and the difference of cubes. Mastering these formulas is...

TermDefinitionExample Perfect CubeA number or expression that is the result of multiplying a number or expression by itself three times.27 is a perfect cube because 3³ = 27. The term 64x⁶ is a perfect cube because (4x²)³ = 64x⁶. Cubic BinomialA polynomial with two terms where the highest degree of the variable is three.x³ - 8 Sum of CubesA cubic binomial that can be written in the form a³ + b³.y³ + 125, which can be written as y³ + 5³. Difference of CubesA cubic binomial that can be written in the form a³ - b³.8x³ - 1, which can be written as (2x)³ - 1³. Irreducible Quadratic FactorA quadratic factor that cannot be factored further using real numbers. The quadratic part of the sum/difference of cubes formula is always irreducible over the reals.In the factorization of x³ + 8 = (x+2)(x²-2x...

Sum of Cubes Formula a³ + b³ = (a + b)(a² - ab + b²) Use this formula when you have two perfect cubes added together. The resulting factors are a binomial and a trinomial. Difference of Cubes Formula a³ - b³ = (a - b)(a² + ab + b²) Use this formula when you have one perfect cube subtracted from another. The resulting factors are a binomial and a trinomial. SOAP Mnemonic for Signs SOAP: Same, Opposite, Always Positive A memory aid for the signs in the formulas. The first sign in the factored form is the **S**ame as the original expression. The second sign is the **O**pposite. The third sign is **A**lways **P**ositive.