Problem 63 - Elementary Algebra

Question

For the following problems, perform the indicated operations. $$ \frac{x^{2}-x-12}{x^{2}-3 x+2} \cdot \frac{x^{2}+3 x-4}{x^{2}-3 x-18} $$

Step-by-Step Solution

Answer

Question: Simplify the algebraic expression: \(\frac{x^2 - x - 12}{x^2 - 3x + 2} \cdot \frac{x^2 + 3x - 4}{x^2 - 3x - 18}\). Answer: \(\frac{(x - 4)(x + 4)}{(x - 2)(x - 6)}\)

Step 1: Factorize numerators and denominators

First, factor the numerators and denominators of both fractions: The first fraction: - Numerator: \(x^2 - x - 12\) = \((x - 4)(x + 3)\) - Denominator: \(x^2 - 3x + 2\) = \((x - 1)(x - 2)\) The second fraction: - Numerator: \(x^2 + 3x - 4\) = \((x + 4)(x - 1)\) - Denominator: \(x^2 - 3x - 18\) = \((x - 6)(x + 3)\) Now, substitute these factors back into the fractions: $$ \frac{(x - 4)(x + 3)}{(x - 1)(x - 2)} \cdot \frac{(x + 4)(x - 1)}{(x - 6)(x + 3)} $$

Step 2: Cancel common factors

Next, cancel out the common factors that appear in both the numerators and denominators of the fractions: - \((x + 3)\) is a common factor in the numerator of the first fraction and the denominator of the second fraction - \((x - 1)\) is a common factor in the denominator of the first fraction and the numerator of the second fraction After canceling the common factors, we get: $$ \frac{(x - 4)}{(x - 2)} \cdot \frac{(x + 4)}{(x - 6)} $$

Step 3: Multiply the simplified fractions

Now, multiply both numerators together followed by both denominators together and then combine them as a single fraction: Numerator: \((x - 4)(x + 4)\) Denominator: \((x - 2)(x - 6)\) The final simplified fraction is: $$ \frac{(x - 4)(x + 4)}{(x - 2)(x - 6)} $$

Key Concepts

Simplifying Rational Expressions

When simplifying rational expressions, such as fractions that contain polynomials in the numerator and denominator, the first step is to factorize these polynomials. Factoring is all about breaking down a larger expression into products of simpler ones. For example, the expression \(x^2 - x - 12\) can be factored into \(x - 4)(x + 3)\).

Once we factorize both the numerator and the denominator, we look for common factors that can be canceled out. This reduces the complexity and often leads to a much simpler expression. Simplifying makes further operations, such as addition, subtraction, multiplication, or division of rational expressions, easier to handle.

Polynomial Operations

Polynomial operations include addition, subtraction, multiplication, and division of polynomials. In the context of rational expressions, after we factorize as a part of simplification, multiplying polynomials is the next step. It's similar to multiplying numbers, but we have to apply the distributive property. For instance, \(x - 4)(x + 4)\) becomes \(x^2 - 16\) after applying the distributive property, which states that each term in the first binomial must be multiplied by each term in the second binomial.

Key polynomial operations can transform a complex fraction to a more manageable form, ready for further evaluation like simplification or cancellation of common factors.

Cancelling Common Factors

Cancelling common factors is an integral part of simplifying rational expressions. After factoring, we observe the numerator and denominator for factors that are identical and divide them out. This is valid because anything divided by itself equals one. It's a bit like reducing fractions to their lowest terms. In the given example, \(x + 3)\) and \(x - 1)\) were identified as common factors and thus, were canceled out from the expression, which simplifies the expression significantly. Canceling common factors not only simplifies your expression but also sometimes reveals further insights into the behavior of the function, such as undefined points or potential holes when graphed.

Multiplying Fractions

The process of multiplying fractions is fairly straightforward: multiply the numerators together and do the same for the denominators, then write the product as a fraction. For polynomial fractions, after canceling the common factors, we multiply the remaining factors as shown in the example. \(x - 4)\) multiplies with \(x + 4)\) to form the numerator and \(x - 2)\) multiplies with \(x - 6)\) to form the denominator of the new, simplified expression. The multiplication of these factors can sometimes result in recognizable patterns, such as the difference of squares—key insights that can make further algebraic manipulations and simplifications more manageable.