Problem 18 - Algebra 2

Question

Simplify each expression. \(\frac{5 t-5}{t^{2}-1}\)

Step-by-Step Solution

Answer

The simplified expression is \(\frac{5}{t+1}\).

Step 1: Factor the Numerator

The numerator of the expression is \(5t - 5\). Notice that 5 is a common factor that can be factored out. Thus, the expression becomes \(5(t - 1)\).

Step 2: Factor the Denominator

The denominator of the expression is \(t^2 - 1\). This is a difference of squares, which can be factored as \((t - 1)(t + 1)\).

Step 3: Write the Factored Expression

After factoring, we substitute back into the original fraction to get \(\frac{5(t - 1)}{(t - 1)(t + 1)}\).

Step 4: Simplify the Expression

Notice that \((t - 1)\) is present in both the numerator and the denominator, thus they can cancel each other out. This simplifies the expression to \(\frac{5}{t + 1}\) as long as \(t eq 1\) to avoid division by zero.

Key Concepts

Factoring Polynomials

Factoring polynomials is all about breaking down a complex expression into simpler components. This process is like taking a big puzzle and finding its smaller parts. For the numerator of our expression, which is \(5t - 5\), we can observe that each term includes a common factor, 5. Hence, we factor out the 5, turning the expression into \(5(t - 1)\).

• This step simplifies the polynomial by recognizing common factors easily.
• Always look for constants or variables that can be factored out across all terms.
• This not only simplifies the expression but makes the next steps easier to manage.

Understanding how to factor polynomials is crucial, as it enables us to simplify and solve algebraic expressions efficiently.

Difference of Squares

The difference of squares is a special technique used to factor expressions where one square is subtracted from another. It's a shortcut that can make simplifying polynomials much easier. For the denominator \(t^2 - 1\), notice how it fits the pattern \(a^2 - b^2\), where \(a = t\) and \(b = 1\). This expression can be rewritten as \((t - 1)(t + 1)\).

• Recognizing these patterns is key to quickly factoring without lengthy calculations.
• Not all polynomials can be factored this way, but the difference of squares is a common scenario.
• Using patterns like this can save time and reduce complex expressions into simpler ones.

Being able to quickly identify and factor a difference of squares streamlines problem-solving and is an essential skill in algebra.

Simplifying Expressions

Simplifying expressions often feels like tidying up mathematical sentences. Once we have our factored expression \(\frac{5(t - 1)}{(t - 1)(t + 1)}\), the goal is to remove common terms. Here, \(t - 1\) appears in both the numerator and the denominator. This term can be cancelled out, resulting in a simplified expression \(\frac{5}{t + 1}\).

• Cancelling like terms makes the expression neater and often easier to evaluate.
• While simplifying, always consider constraints like division by zero, which for this expression means \(t \, eq \, 1\).
• Simplification can often change how the expression behaves, so recheck constraints frequently.

The art of simplifying expressions lies in recognizing such possibilities and making the expression as clear and straightforward as possible.