Factor by any method. See Examples 1–7. 64+(3x+2)³

Verified step by step guidance

1. Recognize that the expression is a sum of two terms: \(64\) and \((3x+2)^3\). Notice that \(64\) is a perfect cube since \$64 = 4^3$.
2. Rewrite the expression as a sum of cubes: \$4^3 + (3x+2)^3$.
3. Recall the sum of cubes factoring formula: \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\).
4. Identify \(a = 4\) and \(b = (3x+2)\), then apply the formula: \((4 + (3x+2)) \left(4^2 - 4(3x+2) + (3x+2)^2\right)\).
5. Simplify the first factor and expand the terms inside the second factor to write the fully factored form.

Key Concepts

Sum of Cubes Formula

The sum of cubes formula states that a³ + b³ = (a + b)(a² - ab + b²). It is used to factor expressions where two terms are each perfect cubes added together. Recognizing the expression as a sum of cubes allows for straightforward factoring.

Identifying Perfect Cubes

To apply the sum of cubes formula, each term must be a perfect cube. For example, 64 is 4³, and (3x + 2)³ is already a cube. Recognizing these cubes is essential to rewrite the expression in a form suitable for factoring.

Polynomial Factoring Techniques

Factoring polynomials involves rewriting expressions as products of simpler polynomials. Techniques include factoring by grouping, special formulas like sum/difference of cubes, and recognizing patterns. Choosing the correct method simplifies solving or further manipulation.