Problem 28 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

The simplified form is \(\frac{a(a^2 + 3)}{2(2a - 1)}\).

Step 1: Factor the Numerators and Denominators

First, factor each potential polynomial in the numerators and denominators. For \(2a^2 + 6\), factor out the greatest common factor: \[2 (a^2 + 3).\]For \(a^2 - a\), factor out \(a\): \[a(a - 1).\]For \(a^3 - a^2\), factor out \(a^2\): \[a^2 (a - 1).\]Lastly, for \(8a - 4\), factor out the greatest common factor: \[4(2a - 1).\]

Step 2: Re-write the Expression with Factored Forms

Substitute the factored forms into the rational expression:\[\frac{2(a^2 + 3)}{a(a - 1)} \cdot \frac{a^2(a - 1)}{4(2a - 1)}.\]

Step 3: Simplify by Cancelling Common Factors

Now, simplify the expression by cancelling out common factors in the numerators and denominators.First, notice the factor \(a - 1\) appears in both fractions and can be cancelled out:\[\frac{2(a^2 + 3)}{a} \cdot \frac{a^2}{4(2a - 1)}.\]Next, cancel the common factor of \(a\) in the denominator of the first fraction and numerator of the second:\[\frac{2(a^2 + 3)}{1} \cdot \frac{a}{4(2a - 1)}.\]

Step 4: Perform Division and Multiply Remaining Factors

Multiply the remaining factors:\[\frac{2(a^2 + 3) imes a}{4(2a - 1)}.\]This simplifies to:\[\frac{2a(a^2 + 3)}{4(2a - 1)}.\]

Step 5: Simplify the Expression

Finally, simplify by combining constants and cleaning the expression. Divide both numerator and denominator by 2:\[\frac{a(a^2 + 3)}{2(2a - 1)}.\]This is the simplest form of the expression.

Key Concepts

Factoring Polynomials

Factoring polynomials is a crucial step when dealing with rational expressions. It involves breaking down a complex polynomial into simpler components called factors. This allows for easier manipulation and simplification.
For example, let's consider the polynomial term in our exercise, like \(2a^2 + 6\). Here, both terms share a common factor of 2. Factoring out this common number, we simplify the expression to \(2(a^2 + 3)\).
Factoring can often reveal common factors in different parts of a rational expression, making it simpler to cancel and simplify further down the line. Understanding how and why to factor can be a powerful tool in making algebraic problems more tractable and less intimidating.

Simplifying Expressions

Simplifying expressions is a process that involves reducing a complex equation or expression into a more basic and manageable form.
In terms of rational expressions, like the one in our example, this involves canceling out common factors between the numerator and the denominator after factoring them fully.
By simplifying, we aim to find the most straightforward form of an expression which allows easier understanding and further manipulations. Recognizing patterns in algebraic expressions also aids simplification, helping in finding equivalent but simpler expressions.

Algebraic Operations

Algebraic operations include addition, subtraction, multiplication, and division of algebraic expressions. For rational expressions, multiplication and division require careful handling of both numerators and denominators.
In our exercise, multiplication is performed by multiplying numerators together and denominators together. First, we multiply the simplified numerators as shown: \(2(a^2 + 3) \cdot a\), resulting in \(2a(a^2 + 3)\). Similarly, the denominators \(a \cdot 4(2a - 1)\) become \(4a(2a - 1)\).
After performing these operations, additional simplifications can be made if there are remaining common factors. Thus, algebraic operations are key to transforming and simplifying complex expressions.

Mathematical Problem Solving

Mathematical problem solving involves a systematic approach to finding solutions to complex problems. This includes understanding the problem, devising a plan, carrying out that plan, and then reviewing the solution.
In our particular exercise, recognizing that the rational expression needs simplification guides the problem-solving strategy. By factoring and simplifying at the beginning, we made the subsequent multiplication and division straightforward.
The final solution involves making sure we have the expression in its simplest form, dividing common factors like in this case, cancelling the remaining factors by dividing by 2 in both numerator and denominator, ensuring the expression is expressed explicitly and simply. Problem solving in mathematics is as much about the process as it is about finding the answer.