Factor each polynomial. See Example 7. (x-4)³+64

Verified step by step guidance

1. Recognize that the expression \( (x-4)^3 + 64 \) is a sum of cubes, since \$64\( can be written as \)4^3\(. So the expression is of the form \)a^3 + b^3\( where \)a = (x-4)\( and \)b = 4$.
2. Recall the sum of cubes factoring formula: \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\).
3. Apply the formula by substituting \(a = (x-4)\) and \(b = 4\) into the factors: first factor is \((x-4 + 4)\) and the second factor is \(( (x-4)^2 - (x-4)(4) + 4^2 )\).
4. Simplify the first factor: \((x-4 + 4)\) simplifies to \(x\). Then expand and simplify the second factor: expand \((x-4)^2\), multiply \((x-4)(4)\), and calculate \$4^2$.
5. Write the fully factored form as the product of the simplified first factor and the simplified second factor.

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 (x - 4)³ + 64 as a sum of cubes allows us to apply this formula directly.

Identifying Perfect Cubes

A perfect cube is a number or expression raised to the third power, such as x³ or 64 (since 64 = 4³). Identifying each term as a perfect cube is essential before applying the sum or difference of cubes formulas in polynomial factoring.

Polynomial Factoring Techniques

Factoring polynomials involves rewriting them as products of simpler polynomials. Techniques include recognizing special patterns like sum/difference of cubes, difference of squares, and factoring by grouping. Mastery of these methods simplifies solving polynomial equations.