Problem 96 - College Algebra with Modeling and Visualization

Question

Factor the expression. \(8-z^{3}\)

Step-by-Step Solution

Answer

The expression \(8 - z^3\) factors to \((2 - z)(4 + 2z + z^2)\).

Step 1: Recognize the Expression as a Difference of Cubes

The expression given is \(8 - z^3\). Recognize that this expression is a difference of cubes, since \(8 = 2^3\), and we can write it as \(2^3 - z^3\).

Step 2: Apply the Difference of Cubes Formula

The formula for factoring a difference of cubes is \(a^3 - b^3 = (a-b)(a^2+ab+b^2)\). Here, \(a = 2\) and \(b = z\).

Step 3: Substitute into the Formula

Substitute \(a = 2\) and \(b = z\) into the formula: \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\).This gives us:\((2 - z)((2)^2 + (2)(z) + (z)^2)\).

Step 4: Simplify the Expression

Simplify the expression:1. \(a^2 = (2)^2 = 4\)2. \(ab = 2z\)3. \(b^2 = z^2\)So, \(a^2 + ab + b^2 = 4 + 2z + z^2\).

Step 5: Write the Final Factored Expression

Combine the results from the previous steps to write the final factored form: \(8 - z^3 = (2 - z)(4 + 2z + z^2)\).

Key Concepts

Factoring

Factoring is a key concept in algebra that involves breaking down an expression into simpler "factors" that, when multiplied together, give you the original expression. In many cases, this involves recognizing patterns or special formulas, like the difference of cubes. For example, with an expression like \(8 - z^3\), recognizing it as a difference of cubes is essential. A difference of cubes formula is an easy way to factor expressions that might seem complex but follow a standard pattern.

• Factoring makes expressions more manageable and helps in simplifying them or solving equations.
• Understanding differences of cubes means spotting cubes and knowing formulas.

To factor \(8 - z^3\), we identify numbers as cubes and apply the cube formula: \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\). This reduces work and helps see the expression's inner structure.

Polynomial Expressions

Polynomial expressions are mathematical phrases that involve variables and coefficients, combined using addition, subtraction, and multiplication. Each part of a polynomial is called a "term," and the highest power of the variable in the expression determines its "degree."

In the expression \(8 - z^3\), we have two terms: \(8\), which is a constant, and \(-z^3\), a cubic term.

• The expression is a quadratic polynomial after factoring, shown as \((2-z)(4 + 2z + z^2)\).
• Factoring involves rewriting the polynomial as a product of simpler polynomials, which can clarify solutions and help with graph sketching or solving equations.

Polynomials like this one often need simplifying or factoring to reveal their behavior and solve related equations. Recognizing the difference of cubes provides a straightforward path to factor and simplify.

Algebra

Algebra is a branch of mathematics dealing with symbols and rules for manipulating those symbols. It forms the foundation for many advanced math topics and involves the solving of equations and the study of functions.

In algebra, understanding how to manipulate expressions, such as the difference of cubes \(8 - z^3\), is crucial. This involves:

• Recognizing patterns and formulas like the difference of cubes formula.
• Applying these formulas to simplify expressions or solve equations.

Algebra uses series of steps to break down complex problems into simpler ones, often leading to solutions that seem elegant or straightforward once properly understood. Factoring, as shown, is a powerful tool in your algebra toolkit, highlighting relationships and solutions within polynomial expressions.