Problem 29 - Intermediate Algebra

Question

Multiply, and then simplify, if possible. See Example 3. $$ \frac{3 t^{2}-t-2}{6 t^{2}-5 t-6} \cdot \frac{4 t^{2}-9}{2 t^{2}+5 t+3} $$

Step-by-Step Solution

Answer

\(\frac{(3t + 2)(t - 1)(2t - 3)}{(2t + 1)(3t - 6)(t + 1)}\)

Step 1: Factor Numerators and Denominators

First, we need to factor each polynomial in the numerators and denominators:- The first numerator is \(3t^2 - t - 2\). This factors to \((3t + 2)(t - 1)\).- The first denominator is \(6t^2 - 5t - 6\). This factors to \((2t + 1)(3t - 6)\).- The second numerator is \(4t^2 - 9\), a difference of squares, which factors to \((2t + 3)(2t - 3)\).- The second denominator is \(2t^2 + 5t + 3\). This factors to \((2t + 3)(t + 1)\).

Step 2: Set Up the Fraction Multiplication

Now substitute the factored forms into the multiplication:\[\frac{(3t + 2)(t - 1)}{(2t + 1)(3t - 6)} \cdot \frac{(2t + 3)(2t - 3)}{(2t + 3)(t + 1)}\]

Step 3: Cancel Common Factors

Identify and cancel any common factors in the numerators and denominators:- The \((2t + 3)\) term appears in both a numerator and a denominator, so it can be cancelled.This simplifies the expression to:\[\frac{(3t + 2)(t - 1)(2t - 3)}{(2t + 1)(3t - 6)(t + 1)}\]

Step 4: Simplify the Expression Further if Possible

Review the remaining factors to see if there are additional simplifications.- No other immediate common factors present.Thus, the simplified expression is:\[\frac{(3t + 2)(t - 1)(2t - 3)}{(2t + 1)(3t - 6)(t + 1)}\]

Key Concepts

Polynomial Factoring

Polynomial factoring involves breaking down a polynomial equation into products of simpler polynomials. Think of it like taking something whole and pulling it apart into pieces that multiply back to the original.
Here's how this works:

• The polynomial equation is split into factors or pieces.
• These pieces are easier to work with, especially for solving problems.

In the exercise, each of the polynomial components such as \(3t^2 - t - 2\) is factored into two simpler expressions \((3t + 2)(t - 1)\).
The final goal is to multiply these smaller equations easily.

Rational Expressions

Rational expressions are fractions where the numerator and/or the denominator are polynomials. Just like normal fractions, you can multiply, divide, add, or subtract them.

When working with rational expressions:

• Treat the numerator and denominator like you would in basic fractions.
• Always look for opportunities to simplify by factoring.

In the original exercise, rational expressions come into play. Both fractions can seem complicated at first, but by factoring, they become easier to manage.
The key is setting up your fractions with these factored forms so you can see simplifications.

Simplifying Fractions

Simplifying is when you make a fraction as simple and straightforward as possible. For rational expressions, simplify by cancelling out common factors.
This is like reducing a normal fraction (like turning \(\frac{4}{8}\) into \(\frac{1}{2}\)).

Here's how to simplify:

• Factor both the numerator and denominator.
• Cancel any common terms (the same part on top and bottom).

In the problem, we reduced the expression by cancelling out \((2t + 3)\) since it appears in both a numerator and denominator.
This process helps the rational expression become its simplest form.