Problem 56 - A Graphical Approach to College Algebra

Question

Factor each sum or difference of cubes completely. $$27 z^{3}+729 y^{3}$$

Step-by-Step Solution

Answer

The factors are \(3(z + 3y)(9z^{2} - 27zy + 81y^{2})\).

Step 1: Identify the Formula

The expression given, \(27z^{3} + 729y^{3}\), is a sum of cubes. We use the sum of cubes formula: \(a^{3} + b^{3} = (a + b)(a^{2} - ab + b^{2})\).

Step 2: Identify 'a' and 'b' in the Expression

Rewrite \(27z^{3}\) and \(729y^{3}\) to match the sum of cubes formula. We have \((3z)^{3} + (9y)^{3}\), so, \(a = 3z\) and \(b = 9y\).

Step 3: Apply the Sum of Cubes Formula

Substitute \(a = 3z\) and \(b = 9y\) into the sum of cubes formula: \((3z + 9y)((3z)^{2} - (3z)(9y) + (9y)^{2})\).

Step 4: Simplify the Expression

First, calculate \((3z)^{2} = 9z^{2}\), \((3z)(9y) = 27zy\), \((9y)^{2} = 81y^{2}\). Thus, the expression becomes: \((3z + 9y)(9z^{2} - 27zy + 81y^{2})\).

Step 5: Factor the Resulting Expression

Factor out the greatest common factor from \((3z + 9y)\) which is 3, and similarly check if the quadratic can be factored further. The expression becomes \(3(z + 3y)(9z^{2} - 27zy + 81y^{2})\). Upon inspection, the quadratic does not factor further over the integers.

Key Concepts

Factoring

Factoring is the process of breaking down an expression into a product of its simplest components or factors. It's like finding out what numbers can be multiplied together to get the original number or expression. When it comes to algebra, factoring helps us simplify polynomials and solve equations more easily.

In our example, we are asked to factor the expression \(27z^{3} + 729y^{3}\). This expression is a sum of cubes, and when we factor it, we rewrite it as a product of two simpler expressions using the formula for the sum of cubes. The sum of cubes formula is a specific tool used for these types of problems and is given by:

• \(a^{3} + b^{3} = (a + b)(a^{2} - ab + b^{2})\)

By recognizing the expression as a sum of cubes, we can effectively use this formula to factor it step by step.

Polynomial Expressions

Polynomial expressions are algebraic expressions that involve sums and differences of terms, usually arranged in descending order by degree. Each term consists of a variable raised to a power, multiplied by a coefficient. In our given problem, \(27z^{3} + 729y^{3}\), we have two terms, each a cube of a particular binomial.

This example is specifically formatted as a sum of cubes, a type of polynomial expression. Recognizing patterns in polynomial expressions, such as sums of cubes or differences of cubes, can greatly simplify the process of factoring them.

When working with polynomial expressions:

• Identify the highest power terms and their coefficients.
• Use known algebraic identities and formulas to simplify or factor these expressions, such as recognizing cubes or squares.
• Apply these formulas to break down the expression into more manageable pieces.

Algebraic Formulas

Algebraic formulas are pre-established identities that help simplify operations involving polynomial expressions. These formulas are foundational tools in algebra that allow for efficient factoring and solving of complex expressions.

The sum of cubes formula is one such useful algebraic formula. It provides a method to break down expressions like \(a^3 + b^3\) into simpler expressions. Using algebraic formulas, particularly those dealing with special products like cubes, empowers us to write lengthy polynomials in a simpler, factored form.

When utilizing algebraic formulas:

• Identify the correct formula to use based on the polynomial's pattern (e.g., sum of cubes, difference of squares).
• Substitute the appropriate values into the formula, as seen with \(a = 3z\) and \(b = 9y\) in our example.
• Simplify step-by-step to ensure clarity and correctness of the final expression.

Learning to apply these formulas accurately is key to mastering more complex algebraic manipulations and solutions.