Problem 23 - Algebra and Trigonometry

Question

\(21-34\) . Perform the multiplication or division and simplify. $$ \frac{x^{2}-x-12}{x^{2}-9} \cdot \frac{3+x}{4-x} $$

Step-by-Step Solution

Answer

The simplified expression is \(-\frac{1}{x-3}\).

Step 1: Factorize the Quadratics

First, factorize the quadratic expressions in the numerators and denominators. The expression in the first fraction's numerator is \(x^2-x-12\). This can be factorized into \((x-4)(x+3)\). The expression in the first fraction's denominator is \(x^2-9\), which is a difference of squares and can be factorized into \((x-3)(x+3)\).

Step 2: Rewrite and Factor the Fractions

Now rewrite the entire expression using the factored forms from Step 1. \[ \frac{(x-4)(x+3)}{(x-3)(x+3)} \cdot \frac{3+x}{4-x} \]Notice that the second fraction \(\frac{3+x}{4-x}\) can be rewritten as \(\frac{x+3}{-(x-4)}\) due to distribution and rearrangement.

Step 3: Cancel Common Factors

Now cancel out the common factors from the numerators and denominators. The common term \((x+3)\) can be canceled out from both fractions. This leaves us with:\[ \frac{(x-4)}{(x-3)} \cdot \frac{1}{-(x-4)} \]

Step 4: Simplify the Expression

Now simplify the expression after canceling. The terms \((x-4)\) in the numerator and denominator cancel out completely, leaving:\[ \frac{1}{-(x-3)} \]Thus, the simplified form is:\[ -\frac{1}{x-3} \]

Key Concepts

Factoring Quadratics

When dealing with algebraic fractions, especially those containing quadratics, it is crucial to first factor these quadratics. Factoring quadratics involves rewriting a quadratic equation in the form of products of binomials. For example, given the quadratic expression \(x^2-x-12\), our task is to find two numbers that multiply to \(-12\) and add up to \(-1\). Here, those numbers are \(-4\) and \(3\), which allows us to factor the expression as \((x-4)(x+3)\).
Another common scenario in factoring quadratics is the difference of squares. This occurs when you have a quadratic in the form \(a^2-b^2\), which can be factored into \((a-b)(a+b)\). In our problem, \(x^2-9\) is an example of a difference of squares, since it can be written as \((x-3)(x+3)\).
By identifying and applying these factoring techniques, you simplify complex fractions and make further operations much easier.

Simplifying Expressions

Simplifying algebraic fractions involves reducing them to their most basic form without changing their value. The key is identifying and canceling out common factors in the numerators and denominators. After factoring, check for factors that appear in both the numerator and the denominator. For instance, in our exercise, both numerator and denominator contain \((x+3)\), which can be canceled out immediately.
Once these common factors are eliminated, you're left with simpler terms. Always ensure that any canceled factors do not cause division by zero in the original context of the problem. Make sure to handle negative signs properly, as they can impact the final expression’s interpretation. In our solution, when simplifying \(\frac{x+3}{4-x}\), we observe it can be rewritten as \(\frac{x+3}{-(x-4)}\), capturing the change in sign essential for proper simplification.

Rational Expressions

Understanding rational expressions is essential in algebra. These are fractions where the numerator and denominator are polynomials. The operation performed on these expressions often involve multiplication, division, addition, or subtraction.
When multiplying rational expressions, like in our exercise, you multiply across numerators and denominators while keeping an eye out for common factors that can be canceled. This simplification makes the product much easier to work with. Always factorise wherever possible before carrying out operations.
Rational expressions are simplified further by understanding restrictions on the variable. Variables that make any denominator zero should be noted, as they are not in the domain of the expression. For instance, in \(\frac{1}{-(x-3)}\), it is important to remember that \(xeq3\) because it would result in division by zero. Such considerations are crucial in ensuring the final expression is both simple and defined correctly.