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.
2. Recall the sum of cubes formula: \(x^3 + y^3 = (x + y)(x^2 - xy + y^2)\).
3. Identify \(x = a+1\) and \(y = 3\) since \(27 = 3^3\).
4. Apply the sum of cubes formula: \((a+1)^3 + 3^3 = ((a+1) + 3)((a+1)^2 - (a+1)(3) + 3^2)\).
5. Simplify the expression: \((a+4)((a+1)^2 - 3(a+1) + 9)\).

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 methods include factoring by grouping, using the difference of squares, and applying special formulas like the sum or difference of cubes.

Sum of Cubes

The sum of cubes is a specific algebraic identity that states a^3 + b^3 can be factored into (a + b)(a^2 - ab + b^2). This identity is crucial when dealing with polynomials that include cubic terms, as it allows for simplification and easier manipulation of the expression. Recognizing this pattern is key to efficiently factoring expressions like (a + 1)^3 + 27.

Binomial Expansion

Binomial expansion refers to the process of expanding expressions that are raised to a power, such as (a + b)^n. The expansion can be achieved using the Binomial Theorem, which provides a formula for calculating the coefficients of the terms in the expansion. Understanding this concept is important for recognizing how to manipulate and factor polynomials that involve binomials raised to powers.