Problem 49 - Algebra for College Students

Question

For Problems 13-50, perform the indicated operations involving rational expressions. Express final answers in simplest form. \(\frac{a^{2}-4 a b+4 b^{2}}{6 a^{2}-4 a b} \cdot \frac{3 a^{2}+5 a b-2 b^{2}}{6 a^{2}+a b-b^{2}} \div \frac{a^{2}-4 b^{2}}{8 a+4 b}\)

Step-by-Step Solution

Answer

The simplest form is \(\frac{2(a - 2b)}{a}\).

Step 1: Factor each expression

First, factor each of the polynomials in the rational expressions.- The numerator of the first fraction is a perfect square trinomial: \(a^2 - 4ab + 4b^2 = (a-2b)^2\).- The denominator of the first fraction can be factored by taking out the common factor: \(6a^2 - 4ab = 2a(3a - 2b)\).- The numerator of the second fraction can be factored using standard trinomial factoring: \(3a^2 + 5ab - 2b^2 = (3a - b)(a + 2b)\).- The denominator of the second fraction can be factored similarly: \(6a^2 + ab - b^2 = (3a - b)(2a + b)\).- The numerator of the third fraction is a difference of squares: \(a^2 - 4b^2 = (a - 2b)(a + 2b)\).- The denominator of the third fraction is \(8a + 4b = 4(2a + b)\).

Step 2: Rewrite the division as multiplication

Revise the problem by rewriting the division operation. When you divide by a fraction, it's equivalent to multiplying by its reciprocal.Your operation becomes:\[\frac{(a-2b)^2}{2a(3a-2b)} \cdot \frac{(3a-b)(a+2b)}{(3a-b)(2a+b)} \cdot \frac{4(2a+b)}{(a-2b)(a+2b)}\]

Step 3: Cancel common factors

Now, look for common factors in the numerators and denominators that can be canceled.- The factor \((a + 2b)\) appears in both the second fraction's numerator and the third fraction's denominator.- The factor \((a - 2b)\) is present in both the first fraction's numerator and the third fraction.- The factor \((3a - b)\) appears once in both the numerator and denominator of the second fraction, so they cancel out as well.- Similarly, \((2a + b)\) is a factor in both the second fraction's denominator and the third fraction's numerator.After canceling all possible factors, the simplified expression is:\[\frac{2(a - 2b)}{a}\]

Step 4: Simplify the remaining expression

The fraction \(\frac{2(a - 2b)}{a}\) is already in a simplified form and cannot be reduced further. Therefore, this is the simplest form of the expression.

Key Concepts

Factoring Polynomials

Factoring polynomials is a crucial skill when working with rational expressions. The goal is to rewrite the polynomial as a product of simpler expressions, known as factors, that multiply together to give the original polynomial. This process helps in simplifying and solving polynomial equations.

When factoring, you may encounter several types of polynomials:

• Perfect Square Trinomials: These can be rewritten in the form \((a-b)^2\). For example, \(a^2 - 4ab + 4b^2 = (a - 2b)^2\).
• Common Factor: Look for terms that can be pulled out from each term of the polynomial. In \(6a^2 - 4ab\), the common factor is \(2a\), giving us \(2a(3a - 2b)\).
• Standard Trinomials: These can often be factored into a pair of binomials. An example is \(3a^2 + 5ab - 2b^2\), which factors to \((3a - b)(a + 2b)\).
• Difference of Squares: A polynomial like \(a^2 - 4b^2\) is a classic case, factored as \((a - 2b)(a + 2b)\).

Factoring is the first step in simplifying rational expressions since it reveals common factors that can be canceled.

Simplifying Fractions

Simplifying fractions involves reducing them to their lowest terms so that both the numerator and the denominator have no common factors other than 1. This is achieved by canceling out any shared factors between them.

Here's how you can simplify fractions effectively:

• Identify Common Factors: Examine the factored form of both the numerator and the denominator. For instance, in the example \(\frac{(a-2b)^2}{2a(3a-2b)}\), identifying common factors \((a-2b)\) helps in simplification.
• Cancel Out Common Factors: Any factors that appear in both the numerator and the denominator can be canceled out. This is due to the property that \(\frac{x}{x} = 1\), given \(x eq 0\).
• Check for Further Simplification: Once the common factors are canceled, you should re-examine the remaining expression to ensure it can't be reduced further.

Simplifying fractions helps make complex expressions more easy to work with and understand.

Multiplication and Division of Fractions

Multiplying and dividing fractions is a straightforward process, but it requires careful attention to ensure accuracy.

Let's break down these operations:

• Multiplication: You multiply the numerators of the fractions to get the new numerator, and the denominators to get the new denominator. Simplify the resulting fraction by canceling common factors.
• Division: Dividing by a fraction involves multiplying by the reciprocal of that fraction. This means you swap the numerator and denominator of the fraction you are dividing by, turning the division into a multiplication problem. For example, \(\frac{x}{y} \div \frac{a}{b} = \frac{x}{y} \cdot \frac{b}{a}\).
• Simplify the Result: After completing the multiplication or division, factor and simplify the fraction as much as possible by canceling any common factors.

This process ensures that the final expression is in its simplest and most understandable form, making calculations easier and minimizing mistakes.