Problem 61 - Elementary and Intermediate Algebra

Question

Factor. \((a-b)^{3}+27\)

Step-by-Step Solution

Answer

The expression is factored as \((a-b+3)(a^2 - 2ab + b^2 - 3a + 3b + 9)\).

Step 1: Identify the Structure

The expression \((a-b)^3 + 27\) fits the form \(x^3 + y^3\), where \(x = (a-b)\) and \(y = 3\), because \(27\) is \(3^3\). We will use the sum of cubes identity to factor it.

Step 2: Recall the Formula for Sum of Cubes

The formula to factor the sum of cubes \(x^3 + y^3\) is \( (x+y)(x^2 - xy + y^2) \). This will be our guide to solve the expression.

Step 3: Substitute Values and Factor

Substitute \(x = (a-b)\) and \(y = 3\) into the sum of cubes formula: \[ (a-b)^3 + 27 = ((a-b) + 3)((a-b)^2 - (a-b) imes 3 + 3^2). \]

Step 4: Simplify Each Part

Calculate and simplify each part of the factored form:1. \((a-b) + 3 = a-b+3\).2. \((a-b)^2 = a^2 - 2ab + b^2\).3. \(-(a-b) \times 3 = -3a + 3b\).4. \(3^2 = 9\).Combine to form:\[(a-b)^3 + 27 = (a-b+3)(a^2 - 2ab + b^2 - 3a + 3b + 9).\]

Step 5: Combine and Final Simplification

The expression inside the second set of parentheses is simplified:\[ a^2 - 2ab + b^2 - 3a + 3b + 9 \] This is the completely factored form of the original expression using the sum of cubes formula:\[(a-b)^3 + 27 = (a-b+3)(a^2 - 2ab + b^2 - 3a + 3b + 9).\]

Key Concepts

Sum of Cubes Formula

The sum of cubes formula is a highly useful tool in algebra. It allows us to factor expressions of the form \(x^3 + y^3\). This is where the sum of two cubes can be expressed as

• \((x + y)\): a simple addition of the cube roots.
• \((x^2 - xy + y^2)\): a quadratic expression.

The complete formula can be written as:\[ x^3 + y^3 = (x + y)(x^2 - xy + y^2) \]This formula simplifies polynomial expressions by breaking down complex cubes into manageable parts. It removes the need for tedious long multiplications or divisions. In our exercise, knowing this formula lets us express \((a-b)^3 + 27\) in a simplified and factored form. It's like having a shortcut for dealing with cubes effectively!

Algebraic Expressions

Algebraic expressions are foundational in mathematics. They consist of numbers, variables, and operators. In the given exercise, the expression \((a-b)^3 + 27\) involves a polynomial in the variable \(a-b\). This is an example of a composite algebraic expression because it includes operations such as subtraction and exponentiation.

• Understand the components: identify numbers, variables, and operators in an expression.
• Know how to restructure: rearranging or regrouping terms often reveals simpler forms or makes complex expressions easier to manage.
• Substitute and evaluate: just like we substituted \(a-b\) for \(x\), this technique helps in using formulas efficiently.

Recognizing these aspects helps in approaching problems systematically and is a critical skill in algebra.

Polynomial Simplification

Polynomial simplification is the process of reducing a polynomial to its simplest form. It involves operations such as combining like terms and applying well-known identities or formulas.
In the context of our exercise, we simplified the expression by:

• Identifying and grouping similar terms: Like terms such as \(-xa, xb\) can be easily spotted and combined.
• Substituting and recalculating: The idea of substituting \(x = (a-b)\) and \(y = 3\) converts complex polynomials into simpler structures.
• Simplifying each segment separately: Calculating squares and products makes it easier to handle individual parts before combining.

By following these approaches, the challenging task of simplifying polynomials becomes manageable, providing clarity to expressions like \((a-b)^3 + 27\). This simplification paves the way for further algebraic manipulations or analyses.