Work each problem. Which of the following is the correct factorization of x^3+8? A. (x+2)^3 B. (x+2)(x^2+2x+4) C. (x+2)(x^2-2x+4) D. (x+2)(x^2-4x+4)

Key Concepts

Sum of Cubes

The expression x^3 + 8 can be recognized as a sum of cubes, since 8 is 2^3. The formula for factoring a sum of cubes is a^3 + b^3 = (a + b)(a^2 - ab + b^2). In this case, a = x and b = 2, which allows us to apply the formula to factor the expression.

Factoring Techniques

Factoring is the process of breaking down an expression into simpler components, or factors, that when multiplied together yield the original expression. Understanding different factoring techniques, such as factoring by grouping or using special formulas like the difference of squares or sum of cubes, is essential for solving polynomial equations effectively.

Polynomial Expressions

A polynomial expression is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. In this case, x^3 + 8 is a polynomial of degree 3. Recognizing the structure of polynomial expressions helps in identifying appropriate methods for simplification and factorization.