Problem 70 - Intermediate Algebra

Question

Perform each operation and simplify, if possible. See Example 9. $$ \frac{4 a^{2}-10 a+6}{a^{4}-3 a^{3}} \div \frac{3-2 a}{2 a^{3}} \cdot \frac{a-3}{2 a-2} $$

Step-by-Step Solution

Answer

Convert division to multiplication, then multiply and simplify.

Step 1: Simplify the Expression

We begin by simplifying each part of the expression separately if possible. The original expression is\[ \frac{4a^2 - 10a + 6}{a^4 - 3a^3} \div \frac{3 - 2a}{2a^3} \cdot \frac{a - 3}{2a - 2}. \]The fraction \( \frac{a - 3}{2a - 2} \) can be simplified by factoring the denominator as \(2(a-1)\), giving:\[ \frac{a - 3}{2(a - 1)}. \]No further simplification is possible for the remaining fractions in this step.

Step 2: Convert Division to Multiplication

When dividing by a fraction, we multiply by its reciprocal. Thus, the expression becomes:\[ \frac{4a^2 - 10a + 6}{a^4 - 3a^3} \cdot \frac{2a^3}{3 - 2a} \cdot \frac{a - 3}{2(a - 1)}. \]

Key Concepts

Simplifying Rational Expressions

Rational expressions are like fractions, but instead of just numbers, they consist of polynomials. To simplify a rational expression, the first step is to factor both the numerator and the denominator, if possible. This process is very much like simplifying numeric fractions, where we find the greatest common factor.

• Identify and factor out any common factors from the numerator and the denominator.
• Cancel out these common factors to simplify the expression.

In the original exercise, for example, the fraction \( \frac{a - 3}{2a - 2} \) was simplified by factoring the denominator to \( 2(a - 1) \). However, since \( a - 3 \) and \( a - 1 \) aren't like terms, no further cancellation was possible.
Remember, simplifying doesn't just make expressions look nicer; it often makes them easier to work with in more complex algebraic operations.

Multiplying and Dividing Fractions

When working with rational expressions, multiplying and dividing are processes similar to those of regular fractions. Understanding these operations will help you navigate through complex algebra problems with ease.
The secret to dividing fractions is to multiply by the reciprocal.
This is why the division in our original exercise, \( \div \frac{3 - 2a}{2a^3} \), turns into multiplication by \( \frac{2a^3}{3 - 2a} \). This way, division is transformed into multiplication, a process that's often simpler.

• For multiplication, simply multiply across the numerators and the denominators and then simplify.
• For division, multiply by the reciprocal of the divisor, meaning you flip the second fraction upside down before proceeding with multiplication.
• The key is to constantly look for simplification opportunities before heavy calculations, saving time and effort.

Factoring Polynomials

Factoring polynomials is a crucial step in simplifying rational expressions. It involves breaking down a polynomial into product terms that, when multiplied, yield the original polynomial.
This is similar to breaking down a number into its prime factors. Starting with identifying and factoring any common terms is essential.

• Look for the greatest common factor (GCF) in each polynomial. For example, in the denominator \( a^4 - 3a^3 \), you can factor out \( a^3 \), leaving \( a^3(a - 3) \).
• Apply other factoring techniques like difference of squares, trinomials, or grouping based on the problem’s structure.

Remember that factoring not only aids in simplification but also is foundational for solving polynomial equations and inequalities.
Skillfully applying these techniques can greatly simplify computations and help avoid errors in algebraic manipulations.