Factor each polynomial. See Example 7. (a+1)³+27

Verified step by step guidance

1. Recognize that the expression \( (a+1)^3 + 27 \) is a sum of cubes because \( 27 = 3^3 \). So, the expression can be written as \( (a+1)^3 + 3^3 \).
2. Recall the sum of cubes factoring formula: \( x^3 + y^3 = (x + y)(x^2 - xy + y^2) \). Here, \( x = a+1 \) and \( y = 3 \).
3. Apply the formula by substituting \( x \) and \( y \): \( (a+1 + 3) \left((a+1)^2 - (a+1)(3) + 3^2\right) \).
4. Simplify the first factor: \( a + 1 + 3 = a + 4 \).
5. Expand and simplify the second factor: \( (a+1)^2 - 3(a+1) + 9 \). This involves expanding \( (a+1)^2 \), distributing \( -3 \), and then combining like terms.

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 perfect cubes are added together. Recognizing the given polynomial as a sum of cubes allows you to apply this formula directly.

Identifying Perfect Cubes

To use the sum of cubes formula, you must identify terms that are perfect cubes. For example, (a+1)³ is a perfect cube, and 27 is 3³. Recognizing these helps in rewriting the expression in the form a³ + b³ for factoring.

Polynomial Factoring Techniques

Factoring polynomials involves rewriting them as products of simpler polynomials. Understanding different factoring methods, such as factoring sums or differences of cubes, is essential for simplifying expressions and solving equations.