Problem 74 - Elementary Algebra

Question

For the following problems, perform the multiplications and divisions. $$ \frac{x^{2}-x-2}{x^{2}-3 x-4} \cdot \frac{-x^{2}+2 x+3}{-4 x-8} $$

Step-by-Step Solution

Answer

Question: Multiply and simplify the following rational expressions: $(\frac{(x^{2}-x-2)}{(x^{2}-3x-4)}) \cdot (\frac{(-x^{2}+2x+3)}{(-4x-8)})$. Answer: The final simplified expression is $\frac{(x-2)(x-1)(x-3)}{4(x-4)(x+2)}$.

Step 1: Factor the expressions

First, we need to factor the expressions in both the numerator and denominator: $$ (x^{2}-x-2) \Rightarrow (x-2)(x+1), (x^{2}-3 x-4) \Rightarrow (x-4)(x+1), (-x^{2}+2 x+3) \Rightarrow -(x-1)(x-3), (-4 x-8) \Rightarrow -4(x+2) $$

Step 2: Simplify common factors

Now, we need to simplify common factors between the numerators and denominators: $$ \frac{(x-2)(x+1)}{(x-4)(x+1)} \cdot \frac{-(x-1)(x-3)}{-4(x+2)} \\ = \frac{x-2}{x-4} \cdot \frac{(x-1)(x-3)}{4(x+2)} $$

Step 3: Perform multiplication

Lastly, we have to multiply the simplified expressions together: $$ \frac{(x-2)}{(x-4)} \cdot \frac{(x-1)(x-3)}{4(x+2)} \\ = \frac{(x-2)(x-1)(x-3)}{4(x-4)(x+2)} $$ So the final result is: $$ \frac{(x-2)(x-1)(x-3)}{4(x-4)(x+2)} $$

Key Concepts

Factoring Polynomials

Understanding how to factor polynomials is essential when trying to simplify complex algebraic expressions. Factoring involves breaking down a polynomial into simpler 'factor' components that, when multiplied together, give back the original polynomial. For example, considering the quadratic expression \(x^2 - x - 2\) from the exercise, we look for two numbers that multiply to -2 and add to -1, which are -2 and +1. Hence, we can rewrite the expression as \(x - 2\) times \(x + 1\).

Factoring is useful because it can reveal common factors in the numerator and denominator of a rational expression, which can then be cancelled out to simplify the expression. While there are several factoring techniques, like taking out the common factor, using the difference of squares, or applying the quadratic formula, the most appropriate method depends on the form of the polynomial. Knowing these techniques allows for efficiency when dealing with polynomial multiplication and division as in the given exercise.

Simplifying Rational Expressions

Simplifying rational expressions is a process that reduces the complexity of fractions that have polynomials in both their numerators and denominators. After factoring the polynomials, as mentioned previously, you can cancel out common factors. It is important to remember that factors can only be cancelled if they are present in both the top and bottom of the fraction.

In the exercise, \(x + 1\) was a common factor that appeared in both the numerator and the denominator of our expression, allowing us to simplify the expression significantly. Note that this is like simplifying the fraction \(4/8\) to \(1/2\) by cancelling the common factor of 4. This process is integral to working with rational expressions, as it makes further arithmetic operations more straightforward and the final expressions much easier to understand and evaluate.

Multiplying Rational Expressions

Multiplying rational expressions is analogous to multiplying fractions; we simply multiply the numerators with each other and the denominators with each other. However, before rushing into the multiplication, it's often beneficial to factor and simplify, as we've done with the exercise provided. This preemptive simplification can drastically reduce the complexity of the problem.

On continuing with our exercise, after simplifying, we multiplied the remaining terms in the numerator and the denominator to reach the final result. Multiplication of rational expressions can lead to longer polynomials in both the numerator and the denominator, but remember to look out for any further simplifications even after you've multiplied, as new common factors can sometimes emerge. Through consistent practice, students can start to recognize patterns and become more adept at both simplifying and multiplying rational expressions.