Problem 93 - College Algebra with Modeling and Visualization

Question

Factor the expression. \(y^{3}+z^{3}\)

Step-by-Step Solution

Answer

The expression factors to \((y+z)(y^2 - yz + z^2)\).

Step 1: Recognize the Sum of Cubes Pattern

The expression given, \(y^3 + z^3\), is a sum of cubes. We can apply the sum of cubes formula, which states: \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\). Here, \(y\) and \(z\) play the roles of \(a\) and \(b\) respectively.

Step 2: Apply the Formula

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

Step 3: Write the Final Factored Expression

The factored form of the expression \(y^3 + z^3\) is \((y + z)(y^2 - yz + z^2)\). Make sure to verify that each binomial and trinomial is correctly factored by expanding if necessary, though for this step we are simply writing the final result from the sum of cubes formula.

Key Concepts

Sum of Cubes

The concept of the Sum of Cubes is an important algebraic identity used to simplify certain polynomial expressions. If you have an expression like \(a^3 + b^3\), you are dealing with a sum of cubes. Recognizing this pattern allows you to use a specific formula to factor the expression efficiently. This formula is:

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

This formula breaks down the sum of cubes into a product of two expressions: a binomial \((a + b)\) and a trinomial \((a^2 - ab + b^2)\). It is particularly useful because directly factoring cubes by hand without this identity can be quite cumbersome. By applying this formula, you can simplify polynomials quickly. For example, in the exercise \(y^3 + z^3\), you apply this by setting \(a = y\) and \(b = z\), resulting in the factored form \((y + z)(y^2 - yz + z^2)\). Exploring such algebraic identities helps deepen understanding of polynomial behaviors.

Polynomial Expressions

Polynomials are algebraic expressions that can consist of variables raised to whole-number exponents and coefficients. Understanding polynomials is crucial in algebra as they form the basis for many functions and equations. Here, polynomial expressions such as \(y^3 + z^3\) are constructed by the sum of monomials, each term being a product of a constant and a power of the variable.
In our exercise, \(y^3 + z^3\) represents a polynomial with two terms, each of which is a cube. The sum of cubes formula is a valuable tool for breaking down such expressions by recognizing patterns and applying appropriate identities. By reorganizing polynomials into products of simpler polynomials (factoring), they become easier to solve or simplify further. Factoring helps in finding roots and understanding the behavior of graphs corresponding to these expressions.

Algebraic Identities

Algebraic identities like the sum of cubes are important tools in mathematics for simplifying and solving expressions. An algebraic identity is a universally true equation that holds for all values of the involved variables. These identities, including the sum of cubes (\(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\)), help in factorization processes.

• They provide shortcuts for manipulating expressions.
• These identities save time and reduce computational complexity.
• They are fundamental in deriving further mathematical results.

When dealing with the sum of cubes, recognizing the pattern lets you use the identity rather than expanding individually. This simplifies the work significantly, especially for larger expressions. Mastery of algebraic identities equips you to tackle a wider range of mathematical problems with confidence and creativity.