Problem 96 - Beginning and Intermediate Algebra

Question

Factor completely. $$216 a^{3}+64 b^{3}$$

Step-by-Step Solution

Answer

The completely factorized form of the given expression is \(2(3a+2b)(36a^2 - 24ab + 16b^2)\).

Step 1: Identify A and B

First, we need to find A and B which are the cube roots of the given terms. For the given expression \(216a^3 + 64b^3\), we have: \(A = (216a^3)^\frac{1}{3} = 6a\) \(B = (64b^3)^\frac{1}{3} = 4b\) Now that we have A and B, we can factorize the expression using the sum of cubes formula.

Step 2: Apply the sum of cubes formula

Plug A and B into the sum of cubes formula: \(A^3 + B^3 = (A + B)(A^2 - AB + B^2)\) \(216a^3 + 64b^3 = (6a + 4b)((6a)^2 - (6a)(4b) + (4b)^2)\)

Step 3: Simplify the factors

Now, we simplify the factors: \((6a + 4b)((6a)^2 - (6a)(4b) + (4b)^2) = (2(3a+2b))(36a^2 - 24ab + 16b^2)\)

Step 4: Final factorized expression

The completely factorized expression is: \(216a^3 + 64b^3 = 2(3a+2b)(36a^2 - 24ab + 16b^2)\) So, the completely factorized form of the given expression is \(2(3a+2b)(36a^2 - 24ab + 16b^2)\).

Key Concepts

Sum of Cubes

When working with algebraic expressions, the sum of cubes is a useful factoring method. It applies to cases where you have an expression in the form of \( A^3 + B^3 \). This special form can be factored using a standard formula:

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

To use this formula effectively, it's essential to first identify the cube roots of the terms in the expression, known as \( A \) and \( B \). For instance, if you have \( 216a^3 + 64b^3 \), the cube roots would be \( 6a \) and \( 4b \) respectively.Once you've determined \( A \) and \( B \), you can substitute them into the formula above to factor the polynomial. This transforms the expression into the product of two factors, simplifying the original polynomial.

Algebraic Expressions

Algebraic expressions consist of terms with variables and coefficients combined using operations like addition, subtraction, multiplication, and division. Terms in an expression can be categorized based on the highest exponent of the variable they contain.The expression \( 216a^3 + 64b^3 \) is a polynomial, specifically consisting of two cubic terms. The coefficients of these terms, 216 and 64, play a crucial role in determining the expression's structure. Interpretation and manipulation of such expressions involve understanding the operations between terms and how they contribute to the overall expression.By breaking down complex expressions into recognizable forms, such as cubes, mathematicians simplify problems and find solutions more readily.

Polynomial Factorization

Polynomial factorization involves breaking down complex algebraic expressions into simpler, multiplied components known as factors. It is a crucial step in simplifying algebraic equations and solving polynomial equations.For instance, the expression \( 216a^3 + 64b^3 \) was factored into \( 2(3a+2b)(36a^2 - 24ab + 16b^2) \). The process began with identifying the cube roots of each term, which facilitated using the sum of cubes formula.By simplifying the factors, we make the expression more manageable and suited for further analysis or solving. Mastery of factorization contributes to a deeper understanding of algebraic expressions and helps in finding roots and solutions to polynomial equations efficiently.