Problem 65 - College Algebra with Modeling and Visualization

Question

Factor the expression completely, if possible. \(64 z^{2}-25 z^{4}\)

Step-by-Step Solution

Answer

The expression factors as \(z^2(5z + 8)(5z - 8)\).

Step 1: Identify Common Terms and Rearrange

The expression is given as \(64 z^2 - 25 z^4\). Start by recognizing that both terms involve powers of \(z\). Rewrite the expression to reflect the order of the powers: \(-25z^4 + 64z^2\).

Step 2: Factor Out the Greatest Common Factor (GCF)

Identify the GCF of the terms. Both terms can be divided by \(z^2\). Factor \(z^2\) out of the expression as follows: \(-25z^4 + 64z^2 = z^2(-25z^2 + 64)\).

Step 3: Recognize the Structure of the Expression

The expression \(-25z^2 + 64\) is in the form of a difference of squares \(a^2 - b^2\), where \(a^2 = 25z^2\) and \(b^2 = 64\).

Step 4: Apply Difference of Squares Formula

Use the difference of squares formula \(a^2 - b^2 = (a + b)(a - b)\). Here, \(a = 5z\) and \(b = 8\). Therefore, \(-25z^2 + 64\) becomes \((5z + 8)(5z - 8)\).

Step 5: Combine the Factors

Combine the factored form from Step 2 and Step 4 to fully factor the original expression. Thus, \(64z^2 - 25z^4 = z^2(5z + 8)(5z - 8)\).

Key Concepts

Difference of Squares

The concept of the difference of squares is a handy tool in polynomial factoring. It helps simplify expressions when the terms fit a particular pattern.

A difference of squares occurs in an expression of the form \(a^2 - b^2\). This can be rewritten using the formula:

• \(a^2 - b^2 = (a + b)(a - b)\).

This pattern involves two terms:

• Both are perfect squares.
• They are subtracted from one another.

In our exercise, we identified \(-25z^2 + 64\) as a difference of squares. Here, \(a^2 = (5z)^2\) and \(b^2 = 8^2\).

Using the formula, this expression factors into \((5z + 8)(5z - 8)\). Recognizing patterns like this can greatly simplify solving polynomial equations.

Greatest Common Factor

Finding the greatest common factor (GCF) is often the first step in simplifying algebraic expressions. The GCF of a set of terms is the largest factor that can divide all the terms without a remainder.

Let's see how we identified the GCF in our exercise:

• We looked at the expression \(64z^2 - 25z^4\) and spotted that both terms include \(z\).
• We realized that \(z^2\) is the largest power of \(z\) present in both terms.

By factoring out \(z^2\), we simplified the expression to \(z^2(-25z^2 + 64)\).

This crucial step not only reduces clutter, making the expression easier to handle, but it also sets the stage for further factoring. Finding the GCF first can make the next steps in solving and simplifying expressions more straightforward.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operators like plus and minus signs. They can being quite simple, or intricate and complex.

In algebra, learning to manipulate these expressions is fundamental. Here's a simple breakdown:

• **Terms:** Individual parts separated by plus or minus signs, such as \(64z^2\) and \(-25z^4\).
• **Variables:** Letters representing numbers, in this case, \(z\).
• **Coefficients:** Numbers in front of variables, here they are 64 and -25.

The exercise given is a polynomial expression, a type found frequently in algebra. Understanding the components of these expressions and how to factor them is vital in mathematics.

Factoring converts products into simpler subsets, revealing insights into the structure of algebraic equations. This enables easier solving, graphing, or other manipulations down the line.