Problem 63 - Elementary and Intermediate Algebra

Question

Factor. \(64-(a+b)^{3}\)

Step-by-Step Solution

Answer

The factored form of \(64 - (a+b)^3\) is \((4-a-b)(a^2 + 4a + 2ab + 4b + b^2 + 16)\).

Step 1: Recognize the Expression Form

The expression given is of the form \(64 - (a+b)^3\). Notice that 64 is a cube (since \(64 = 4^3\)), which means we have a difference of two cubes: \(4^3 - (a+b)^3\). This fits the formula \(A^3 - B^3\) where \(A = 4\) and \(B = a+b\).

Step 2: Recall the Formula for Difference of Cubes

The formula for the difference of cubes is \(A^3 - B^3 = (A - B)(A^2 + AB + B^2)\). We will use this formula to factor the expression \(4^3 - (a+b)^3\).

Step 3: Substitute and Factor Using the Formula

Substitute \(A = 4\) and \(B = a + b\) into the difference of cubes formula: \((4 - (a+b))(4^2 + 4(a+b) + (a+b)^2)\).

Step 4: Simplify the Factorized Form

Simplify each part: 1. The first term in the factorization is \(4 - (a+b)\).2. The second term is the sum of: - \(4^2 = 16\) - \(4(a+b) = 4a + 4b\) - \((a+b)^2 = a^2 + 2ab + b^2\) Together, the second term becomes \(16 + 4a + 4b + a^2 + 2ab + b^2\).

Step 5: Write the Final Factorized Expression

Combine the expressions obtained into the fully factorized form: \((4 - a - b)(a^2 + 4a + 2ab + 4b + b^2 + 16)\).

Key Concepts

Difference of Cubes

The concept of the difference of cubes is a crucial one in algebra for factoring complex polynomials. It involves expressions that take the form \(A^3 - B^3\), where both numbers are perfect cubes. To factor this completely, there's a specific formula we use:

• Formula: \(A^3 - B^3 = (A - B)(A^2 + AB + B^2)\)

Let's understand each component:

• First Term: \(A - B\) represents the difference of the bases.
• Second Term: \(A^2 + AB + B^2\) is a trinomial that helps complete the factorization.

Using this formula helps in breaking down polynomials into simpler, more manageable parts. This approach doesn't just reveal the factors but also makes the expression easier to work with in subsequent algebraic operations.

Algebraic Expressions

Algebraic expressions are a fundamental concept in mathematics, consisting of numbers, variables, and operations. In the expression \(64 - (a+b)^3\), we are dealing with a combination of a constant and a binomial raised to a power. Here's a breakdown:

• Numbers: These are constants like 64, which do not change.
• Variables: Symbols (such as \(a\) and \(b\)) that stand for unknown values and can vary.
• Operations: Includes addition, subtraction, and exponentiation.

The power of algebraic expressions lies in their ability to represent mathematical situations flexibly. They can be manipulated, evaluated, and used to model real-world scenarios easily. Understanding how to interpret and manipulate algebraic expressions is key to solving more complex polynomial equations.

Polynomial Simplification

When we say polynomial simplification, we are referring to reducing a polynomial expression to its simplest form. This is essential for making expressions less cumbersome and more straightforward to analyze. In our exercise, the expression \((4 - (a+b))(4^2 + 4(a+b) + (a+b)^2)\) was simplified as follows:

• Simplify Each Term: Calculate and combine like terms.

For example:

• Calculating \(4^2\): The square of 4 is 16.
• Expanding \(4(a+b)\): Distribute the 4 to both \(a\) and \(b\).
• Expanding \((a+b)^2\): Use the formula \((a+b)^2 = a^2 + 2ab + b^2\).

By combining these, the expression becomes \((4 - a - b)(a^2 + 4a + 2ab + 4b + b^2 + 16)\). This streamlined process of polynomial simplification helps in both solving equations and understanding the structure of expressions.