Problem 76 - College Algebra with Modeling and Visualization

Question

Factor the expression completely, if possible. \(64-(t-3)^{2}\)

Step-by-Step Solution

Answer

The expression \(64 - (t-3)^2\) factors to \((11+t)(11-t)\).

Step 1: Recognize the Structure of the Expression

We start by identifying the structure of the given expression: \(64 - (t-3)^2\). This expression fits the form of a difference of squares, \(a^2 - b^2\). Here, \(a^2 = 64\) and \(b^2 = (t-3)^2\).

Step 2: Express as a Difference of Squares

Now, we express both parts of the expression as squares: \(64 = 8^2\) and \((t-3)^2\) remains as is. Thus, the expression becomes \(8^2 - (t-3)^2\).

Step 3: Apply the Difference of Squares Formula

The difference of squares formula states: \(a^2 - b^2 = (a+b)(a-b)\). Applying this to our expression we get: \( (8 + (t-3))(8 - (t-3)) \).

Step 4: Simplify the Expressions

Simplify the terms within the parentheses: \((8 + (t-3)) = (11 + t)\) and \((8 - (t-3)) = (11 - t)\).

Step 5: Write the Final Factored Form

Combine the simplified expressions to write the completely factored form of the original expression: \((11 + t)(11 - t)\).

Key Concepts

Factoring Expressions

Factoring expressions is a crucial skill in college algebra. It involves breaking down a complex expression into simpler pieces or factors. These factors, when multiplied together, recreate the original expression. This process simplifies solving equations and helps in simplifying algebraic expressions to make them more manageable.

To factor an expression:

• Identify common factors in all terms.
• Group and factor by regrouping terms whenever possible.
• Use special factoring formulas like the difference of squares, perfect square trinomials, and sum or difference of cubes.

Recognizing these patterns in expressions allows you to simplify and solve problems more effectively. The solution to many algebraic problems requires the recognition and manipulation of these factoring techniques to arrive at simpler, solvable forms.

Difference of Squares

The difference of squares is a specific factoring technique used in algebra when dealing with expressions of the form \(a^2 - b^2\). This is a common pattern that can be factored into two binomials. Understanding and applying the difference of squares formula is a powerful tool:

\[ a^2 - b^2 = (a+b)(a-b) \]

This method plays a critical role in simplifying and solving various algebraic equations. To use the difference of squares approach, the expression must fit into the structure of two perfect squares subtracted from each other, as in the exercise provided where the expression is \(64 - (t-3)^2\).

By identifying \(a^2 = 8^2\) and \(b^2 = (t-3)^2\), the expression can be rewritten and factored. This demonstrates how recognizing the difference of squares pattern can simplify algebraic challenges effectively. It's a must-know formula for all algebra students.

Quadratic Expressions

Quadratic expressions are polynomials of degree two, generally expressed in the form \(ax^2 + bx + c\). These expressions can often be factored or simplified using various techniques, including the difference of squares and other factorization methods.

Recognizing a quadratic expression is straightforward as it typically involves a term with \(x^2\). The key to factor these successfully:

• Look for patterns such as perfect square trinomials and difference of squares.
• Try different factorization strategies like splitting the middle term.
• Practice makes spotting these patterns easier over time.

For example, when simplifying the expression \(64 - (t-3)^2\), although not a standard quadratic like \(ax^2 + bx + c\), it can be perceived as quadratic after recognizing the underlying binomial squared form. Working through such problems enriches your understanding of polynomial structures and prepares you for more complex algebraic equations.