Problem 47 - Algebra 2

Question

The product of \(\frac{x^{2}-11 x+28}{x^{2}-2 x-35}\) and a second rational expression \(\frac{f(x)}{g(x)}\) is \(\frac{x+4}{x+5}\) What is the second rational expression? Show each step of your work.

Step-by-Step Solution

Answer

The second rational expression is \(\frac{x+4}{x-4}\)

Step 1: Simplifying the given rational expression

We are given the rational expression \(\frac{x^{2}-11 x+28}{x^{2}-2 x-35}\). To simplify it, we need to factorize the numerator and the denominator. The numerator, \(x^{2}-11 x+28\) can be factorised into \((x-4)(x-7)\) and the denominator, \(x^{2}-2 x-35 = (x-7)(x+5)\). Therefore, the given expression simplifies to \(\frac{x-4}{x+5}\)

Step 2: Finding the second rational expression

Now, we are told that \(\frac{x-4}{x+5}\) times the second expression gives \(\frac{x+4}{x+5}\). So the second expression, \(\frac{f(x)}{g(x)}\) is equal to \(\frac{\frac{x+4}{x+5}}{\frac{x-4}{x+5}}\). The \(x+5\) terms cancel out and we are left with \(\frac{x+4}{x-4}\) as our second rational expression.

Key Concepts

Factoring Polynomials

Understanding how to factor polynomials is a fundamental skill in algebra. It's like breaking down a complicated puzzle into simpler pieces. Polynomials are expressions that consist of variables and constants, combined using addition, subtraction, and multiplication. Factoring involves rewriting these expressions as a product of simpler polynomials.
For example, in the polynomial:

• \(x^2 - 11x + 28\), we look for two numbers that multiply to 28 (the constant term) and add to -11 (the coefficient of the linear term, \(x\)).
• These numbers are -4 and -7, so we can express \(x^2 - 11x + 28\) as \((x - 4)(x - 7)\).

Similarly, we factor the denominator \(x^2 - 2x - 35\) into \((x - 7)(x + 5)\) by finding numbers that multiply to -35 and add to -2.
Factoring allows us to simplify expressions and solve them more easily in algebraic operations.

Simplifying Rational Expressions

Simplifying rational expressions is removing any common factors from the numerator and the denominator. This involves first factoring both the top and bottom parts of the expression.
In the given example:

• The original expression \(\frac{x^2 - 11x + 28}{x^2 - 2x - 35}\) can be simplified by factoring it to \(\frac{(x-4)(x-7)}{(x-7)(x+5)}\).

Notice the \((x-7)\) in both the numerator and the denominator.
By canceling this common factor, we simplify the expression to \(\frac{x-4}{x+5}\).
It's important to remember that simplification can only occur when the terms are facts, not when they are added or subtracted. Simplifying makes expressions easier to understand and manipulate, especially in more complex problems.

Multiplying Rational Expressions

Multiplying rational expressions involves multiplying the numerators together and the denominators together. It's straightforward as long as we simplify each expression first.
Given that multiplying involves fractions, it often requires a step of simplification before and after multiplication.
In our example, we start with an already simplified expression \(\frac{x-4}{x+5}\), which we know needs to be multiplied by another unknown rational expression to equal \(\frac{x+4}{x+5}\).
By considering the product and simplifying it appropriately, we find the second expression \(\frac{f(x)}{g(x)}\).
So the operation goes as follows: divide the target expression \(\frac{x+4}{x+5}\) by the given expression \(\frac{x-4}{x+5}\). This simplifies to \(\frac{x+4}{x-4}\).
Multiplying rational expressions is handy because it allows for flexibility in solving polynomial equations and altering expression forms.