Factor each polynomial completely. See Example 6.8t³ + 125

Verified step by step guidance

1. Identify the polynomial as a sum of cubes: \$8t^3 + 125\( can be rewritten as \)(2t)^3 + 5^3$.
2. Recall the formula for factoring a sum of cubes: \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\).
3. Assign \(a = 2t\) and \(b = 5\) based on the expression \((2t)^3 + 5^3\).
4. Apply the sum of cubes formula: Substitute \(a = 2t\) and \(b = 5\) into \((a + b)(a^2 - ab + b^2)\).
5. Simplify the expression: Calculate \((2t + 5)((2t)^2 - (2t)(5) + 5^2)\) to complete the factorization.

Key Concepts

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of its simpler components, or factors. This process is essential for simplifying expressions and solving equations. Common methods include factoring out the greatest common factor, using special products like the difference of squares, and applying techniques for cubic polynomials.

Sum of Cubes

The expression 8t³ + 125 is a sum of cubes, as it can be rewritten as (2t)³ + 5³. The sum of cubes can be factored using the formula a³ + b³ = (a + b)(a² - ab + b²). Recognizing this pattern is crucial for efficiently factoring such expressions.

Cubic Polynomial Factorization

Cubic polynomials can often be factored into linear and quadratic factors. In the case of a sum of cubes, the factorization leads to a linear factor and a quadratic factor. Understanding how to manipulate and apply these factorizations is key to solving polynomial equations and simplifying expressions.