Solve equations with sums and differences of cubes

Learning Objectives Identify expressions that are sums or differences of cubes. Recall and accurately apply the sum of cubes factoring formula. Recall and accurately apply the difference of cubes factoring formula. Solve polynomial equations by factoring them as a sum or difference of cubes. Find all real and complex roots of cubic equations of this form. Verify the solutions to a cubic equation by substitution. How can we find the exact side length of a cube if we know its volume is 8 cubic units more than the volume of another, unknown cube? 🧊 This tutorial focuses on a powerful factoring technique for a specific type of polynomial: sums and differences of cubes. Mastering this skill is crucial for solving higher-degree equations without a calculator and provides a deepe...

TermDefinitionExample Perfect CubeA number or expression that is the result of multiplying a number or expression by itself three times.64 is a perfect cube because 4³ = 64. Similarly, 27x⁶ is a perfect cube because (3x²)³ = 27x⁶. Sum of CubesA binomial expression where two perfect cube terms are added together, in the form a³ + b³.x³ + 8, which can be written as x³ + 2³. Difference of CubesA binomial expression where one perfect cube term is subtracted from another, in the form a³ - b³.y³ - 125, which can be written as y³ - 5³. Root of an EquationA value for a variable that makes an equation true. For a polynomial equation P(x) = 0, the roots are the values of x for which the polynomial evaluates to zero.For the equation x³ - 8 = 0, x = 2 is a root because 2³ - 8 = 8 - 8 = 0. Irreducible...

Sum of Cubes Formula a³ + b³ = (a + b)(a² - ab + b²) Use this formula to factor any binomial that is the sum of two perfect cubes. Identify 'a' (the cube root of the first term) and 'b' (the cube root of the second term) and substitute them into the formula. Difference of Cubes Formula a³ - b³ = (a - b)(a² + ab + b²) Use this formula to factor any binomial that is the difference of two perfect cubes. Identify 'a' and 'b' and substitute them into the formula. A helpful mnemonic for the signs is SOAP: Same, Opposite, Always Positive.