In Exercises 57–64, factor using the formula for the sum or difference of two cubes. 8x^3+125

Verified step by step guidance

1. Recognize that the given expression, 8x^3 + 125, is a sum of two cubes. Recall the formula for factoring the sum of two cubes: a^3 + b^3 = (a + b)(a^2 - ab + b^2).
2. Identify the cube roots of each term. For 8x^3, the cube root is 2x, and for 125, the cube root is 5.
3. Substitute the values of a and b into the formula. Here, a = 2x and b = 5, so the expression becomes (2x + 5)((2x)^2 - (2x)(5) + (5)^2).
4. Simplify each term in the second factor. (2x)^2 becomes 4x^2, (2x)(5) becomes 10x, and (5)^2 becomes 25.
5. Write the fully factored form of the expression: (2x + 5)(4x^2 - 10x + 25).

Key Concepts

Sum of Cubes Formula

The sum of cubes formula states that a^3 + b^3 can be factored as (a + b)(a^2 - ab + b^2). This formula is essential for factoring expressions where two terms are perfect cubes added together, allowing for simplification and solving of polynomial equations.

Identifying Perfect Cubes

To apply the sum of cubes formula, it's crucial to recognize perfect cubes. A perfect cube is a number or expression that can be expressed as the cube of another number or expression, such as 8 (2^3) and 125 (5^3). Identifying these helps in correctly applying the factoring formula.

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of simpler polynomials. This process is fundamental in algebra as it simplifies expressions and makes it easier to solve equations. Understanding how to factor using specific formulas, like the sum of cubes, is a key skill in algebra.