Problem 21 - College Algebra with Corequisite Support

Question

For the following exercises, multiply the rational expressions and express the product in simplest form. \(\frac{t^{2}-1}{t^{2}+4 t+3} \cdot \frac{t^{2}+2 t-15}{t^{2}-4 t+3}\)

Step-by-Step Solution

Answer

The simplified product is \(\frac{t+5}{t+3}\).

Step 1: Factor the Numerators

First, factor the numerators of both rational expressions. For the expression \(t^2 - 1\), recognize it as a difference of squares, which can be factored as \((t-1)(t+1)\). For \(t^2 + 2t - 15\), find two numbers that multiply to -15 and add to 2; these are -3 and 5, so it factors to \((t-3)(t+5)\).

Step 2: Factor the Denominators

Next, factor the denominators. For the first denominator \(t^2 + 4t + 3\), find two numbers that multiply to 3 and add to 4; these are 1 and 3, giving the factorization \((t+1)(t+3)\). For the second denominator \(t^2 - 4t + 3\), find two numbers that multiply to 3 and add to -4; these are -1 and -3, so it factors to \((t-1)(t-3)\).

Step 3: Write the Rational Expressions in Factored Form

Re-write the entire expression using the factored forms: \[\frac{(t-1)(t+1)}{(t+1)(t+3)} \cdot \frac{(t-3)(t+5)}{(t-1)(t-3)}\]

Step 4: Cancel Common Factors

Look for and cancel out common factors in the numerators and denominators. Observe that \((t-1), (t+1), (t-3)\) are common factors that can be canceled out:\[\frac{1}{(t+3)} \cdot (t+5)\]

Step 5: Complete Multiplication and Simplify

Finally, multiply the remaining terms: \[\frac{1 imes (t+5)}{1 imes (t+3)}\]Simplifying the expression, you get \[\frac{t+5}{t+3}\].

Key Concepts

Factoring

Factoring is a crucial skill when dealing with rational expressions, as it simplifies the process of multiplication and simplification. It's the method of breaking down complex expressions into simpler, more manageable products. In this particular exercise, we start by factoring both the numerators and denominators of the expressions.

• Factoring the Numerators: The first numerator is a difference of squares, with the expression being \(t^2 - 1\). The difference of squares is a special form that factors into \((t-1)(t+1)\).
• The second numerator, \(t^2 + 2t - 15\), requires finding two numbers that multiply to \(-15\) and add to 2. These numbers are \(-3\) and 5, which gives the factorization \((t-3)(t+5)\).
• Factoring the Denominators: For the first denominator \(t^2 + 4t + 3\), determine two numbers that multiply to 3 and add to 4, which are 1 and 3, resulting in \((t+1)(t+3)\).
• The second denominator \(t^2 - 4t + 3\) factors into \((t-1)(t-3)\) by finding numbers that multiply to 3 and add to \(-4\), namely \(-1\) and \(-3\).

Factoring helps set the stage for multiplication and simplification, making both processes easier and more efficient.

Multiplication of Rational Expressions

Multiplying rational expressions involves multiplying the numerators together and the denominators together, similar to how you multiply fractions. Once each part is factored, the multiplication becomes straightforward.

• First, re-write the rational expressions in their factored form. This gives us \(\frac{(t-1)(t+1)}{(t+1)(t+3)} \cdot \frac{(t-3)(t+5)}{(t-1)(t-3)}\).
• Next, combine the numerators and the denominators: Now, you're dealing with the product \((t-1)(t+1)(t-3)(t+5)\) over \((t+1)(t+3)(t-1)(t-3)\).

This step lays important foundational work for the simplification by making the common factors clear and easier to cancel out.

Simplification of Expressions

Simplification is the process of cancelling out common factors across the numerator and denominator. This reduces the expression to its simplest form.

• After writing the product, identify and cancel common factors present in both the numerator and the denominator. In our case, \((t-1), (t+1),\) and \((t-3)\) appear in both parts and can be cancelled.
• This leaves us with \(\frac{1}{(t+3)} \times (t+5)\).
• Finally, multiply the simplified terms to reach the final answer: \(\frac{t+5}{t+3}\).

This entire process of identifying and cancelling like terms ensures the result is the most reduced version of the original expression.