Problem 38 - Algebra for College Students

Question

For Problems 13-50, perform the indicated operations involving rational expressions. Express final answers in simplest form. \(\frac{2 x^{4}+x^{2}-3}{2 x^{4}+5 x^{2}+2} \cdot \frac{3 x^{4}+10 x^{2}+8}{3 x^{4}+x^{2}-4}\)

Step-by-Step Solution

Answer

The expression simplifies to \(\frac{(2x^2 - 1)(x^2 + 3)}{(2x^2 + 1)(x^2 - 1)}\).

Step 1: Factorize each Polynomial

Start by looking for common factors or use factoring techniques like grouping or finding roots. Let's factorize the numerators and the denominators:Numerator 1: \(2x^4 + x^2 - 3\) can be written as \((2x^2 - 1)(x^2 + 3)\).Denominator 1: \(2x^4 + 5x^2 + 2\) can be written as \((2x^2 + 1)(x^2 + 2)\).Numerator 2: \(3x^4 + 10x^2 + 8\) can be written as \((3x^2 + 4)(x^2 + 2)\).Denominator 2: \(3x^4 + x^2 - 4\) can be written as \((3x^2 + 4)(x^2 - 1)\).

Step 2: Arrange in a Single Fraction

Now that the expressions are factorized, arrange them into a single fraction to see common terms:\[\frac{(2x^2 - 1)(x^2 + 3)(3x^2 + 4)(x^2 + 2)}{(2x^2 + 1)(x^2 + 2)(3x^2 + 4)(x^2 - 1)}\]

Step 3: Cancel Out Common Terms

Identify and cancel out any common factors from the numerator and the denominator:- \((3x^2 + 4)\) can be cancelled from both the numerator and the denominator.- \((x^2 + 2)\) can also be cancelled from both the numerator and the denominator.After canceling, the expression simplifies to:\[\frac{(2x^2 - 1)(x^2 + 3)}{(2x^2 + 1)(x^2 - 1)}\]

Step 4: Express the Simplified Form

Now express the result in its simplest form. Since there can't be any further simplifications by factoring, the simplified form of the expression is:\(\frac{(2x^2 - 1)(x^2 + 3)}{(2x^2 + 1)(x^2 - 1)}\)

Key Concepts

Factoring Polynomials

When dealing with rational expressions, a crucial step is factoring the polynomials involved. Factoring transforms a complex polynomial into simpler multiple factors, which are easier to manipulate. Here are some popular techniques for factoring:
- **Common Factor Extraction**: Check if there is a number or variable that divides all terms of the polynomial. Extract this as a common factor.- **Quadratic Form Factorization**: For polynomials that resemble quadratic equations, use techniques like finding roots or applying the quadratic formula.- **Special Identities**: Use identities like difference of squares or sum of squares where applicable.
In our exercise, each polynomial, such as the numerator \(2x^4 + x^2 - 3\), is rewritten using these techniques. For example, the polynomial is separated into factors like \((2x^2 - 1)(x^2 + 3)\). Ensuring correct factorization simplifies the subsequent operations.

Simplifying Fractions

After factoring, the next logical step is simplifying the fraction. In rational expression operations, simplification corresponds to arranging a fraction in a form that's easiest to work with and interpret.
- **Single Fraction Formation**: When multiplying rational expressions, combine them into a single fraction by multiplying numerators together and denominators together.- **Identify Common Factors**: Once factorization is done, review both numerators and denominators to locate terms that can potentially be reduced or simplified.
Following our exercise example, the terms from the separate fractions are combined. This results in a new expression: \[\frac{(2x^2 - 1)(x^2 + 3)(3x^2 + 4)(x^2 + 2)}{(2x^2 + 1)(x^2 + 2)(3x^2 + 4)(x^2 - 1)}\]Establishing this single fraction makes it straightforward to proceed with further simplifications.

Canceling Common Factors

For effective simplification of rational expressions, canceling common factors is indispensable. This step guarantees the expression is presented in its simplest form.
- **Recognize Common Terms**: Once in a single fraction, compare factors from the numerator and denominator. Identify factors that appear on both levels.- **Cancel Out Identical Factors**: Reduce the fraction by canceling these shared factors.
In our problem, we spotted \((3x^2 + 4)\) and \((x^2 + 2)\) in both the numerator and denominator. These were canceled out, rendering a much cleaner expression as:\[\frac{(2x^2 - 1)(x^2 + 3)}{(2x^2 + 1)(x^2 - 1)}\]This streamlined approach not only makes the expression more elegant but simplifies any further calculations or evaluations required.