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. Recall the sum of cubes formula: \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\).
2. Identify \(a\) and \(b\) in the expression. Here, \(a = x - 4\) and \(b = 4\) because \(64 = 4^3\).
3. Apply the sum of cubes formula: Substitute \(a = x - 4\) and \(b = 4\) into \((a + b)(a^2 - ab + b^2)\).
4. Calculate \(a + b = (x - 4) + 4 = x\).
5. Calculate \(a^2 - ab + b^2 = (x - 4)^2 - (x - 4)(4) + 4^2\) and simplify each term.

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, solving equations, and understanding the polynomial's roots. Common techniques include factoring out the greatest common factor, using special products, and applying methods like grouping or synthetic division.

Sum of Cubes

The expression (x - 4)^3 + 64 represents a sum of cubes, which can be factored using the formula a^3 + b^3 = (a + b)(a^2 - ab + b^2). In this case, a is (x - 4) and b is 4, allowing us to apply this formula to simplify the expression. Recognizing this pattern is crucial for efficient factoring.

Polynomial Identities

Polynomial identities are equations that hold true for all values of the variable involved. Familiarity with these identities, such as the difference of squares, perfect square trinomials, and the sum/difference of cubes, is vital for factoring polynomials effectively. Understanding these identities helps in recognizing patterns and applying the correct factoring techniques.