Problem 90 - Algebra A Combined Function

Question

Factor each expression completely. $$ (y-6)^{2}-z^{2} $$

Step-by-Step Solution

Answer

The factored form is \(((y-6)-z)((y-6)+z)\).

Step 1: Identify the Formula

The expression \((y-6)^2 - z^2\) can be recognized as a difference of squares. The difference of squares formula is \(a^2-b^2 = (a-b)(a+b)\). Identify \(a\) and \(b\) in the expression.

Step 2: Define Terms as Squares

Rewrite \((y-6)^2\) and \(z^2\) for clarity. Here, \(a = (y-6)\) and \(b = z\), since \((y-6)^2\) is \((a)^2\) and \(z^2\) is \((b)^2\).

Step 3: Apply the Difference of Squares Formula

Use the formula \(a^2-b^2 = (a-b)(a+b)\) on the expression. Substitute \(a\) and \(b\) with \((y-6)\) and \(z\), respectively. This gives \((y-6)-z\) and \((y-6)+z\).

Step 4: Write the Factored Expression

Combine the results into a complete factored expression. The fully factored form of \((y-6)^2 - z^2\) is \(((y-6)-z)((y-6)+z)\).

Step 5: Simplify Each Factor

Further simplify the expression if possible, but since \((y-6)-z\) and \((y-6)+z\) cannot be factored further, we leave the expression as is.

Key Concepts

Factoring Expressions

Factoring expressions is a key technique in algebra that simplifies expressions to make them easier to handle and solve. In the example given, the expression \[ (y-6)^2 - z^2 \]is an instance of the difference of squares, a particular type of expression. The main goal of factoring such expressions is to break them down into products of simpler expressions. This process makes them much easier to manipulate, whether we're solving equations or working with more complex algebraic structures.

Here’s a simple way to think of factoring

• Look for an algebraic identity that matches the expression.
• Break down the terms according to this pattern.
• Simplify or rearrange the terms if possible.

Practicing different scenarios through problems like these reinforces the idea that factoring is about identifying patterns and using them to simplify mathematics.

Algebraic Identities

Algebraic identities are pre-proven expressions that can help simplify calculations. They act as handy shortcuts in solving algebra problems. One such identity is the difference of squares, which is:\[ a^2 - b^2 = (a-b)(a+b) \]These identities are like special formulas that allow us to factor expressions easily and correctly.

You can apply this identity directly when you see a squared term subtracted by another squared term. In our exercise:

• Identify the terms \((y-6)^2\) and \(z^2\) as squared terms.
• Recognize this pattern matches the formula \(a^2-b^2\).
• Express them in the form of \((a-b)(a+b)\).

Understanding and memorizing these identities can significantly speed up the problem-solving process, turning complex expressions into straightforward solutions.

Variable Substitution

Variable substitution is a useful strategy in algebra to simplify complicated expressions. This involves replacing a variable or expression with another to make the equation or problem easier to solve or understand. In our case with \((y-6)^2 - z^2\), we substitute:

• Let \(a = (y-6)\)
• Let \(b = z\)

Thus, our expression takes the form \(a^2 - b^2\), which is much simpler to interpret using the difference of squares formula.

By substituting these variables, the expression's complicated appearance is reduced to a neat application of a known identity. It is a strategy especially beneficial in solving higher-order algebra problems, allowing for a streamlined computation process.