Problem 90
Question
In place of each question mark in Exercises \(75-92,\) put an expression that will make the rational expressions equivalent. $$\frac{x-3}{x^{2}-9}=\frac{1}{?}$$
Step-by-Step Solution
Verified Answer
The missing expression is \( x + 3 \).
1Step 1: Factor the Denominator
Factor the denominator of the given expression. The denominator is \(x^2-9\), which is a difference of squares. Factoring this, we get: \[ x^2 - 9 = (x - 3)(x + 3) \]
2Step 2: Simplify the Rational Expression
Rewrite the original rational expression with the factorized denominator: \[ \frac{x-3}{x^2-9} = \frac{x-3}{(x-3)(x+3)} \]
3Step 3: Cancel Common Factors
Cancel the common factor \(x-3\) in the numerator and the denominator: \[ \frac{x-3}{(x-3)(x+3)} = \frac{1}{x+3} \]
4Step 4: Identify Missing Expression
From the simplified expression \( \frac{1}{x+3} \), it's clear that the missing expression is \(x+3\). Therefore, \[ \frac{x-3}{x^2-9} = \frac{1}{x+3} \]
Key Concepts
Rational ExpressionsDifference of SquaresCanceling Common Factors
Rational Expressions
Rational expressions are quotients of polynomials. They are similar to fractions but instead of just numbers in the numerator and denominator, they have polynomials. To work with rational expressions, we generally aim to simplify them. This means reducing them to their simplest form.
For instance, if we have a rational expression like \(\frac{x-3}{x^2-9}\), we want to simplify it as much as possible. Simplification often involves factoring polynomials in the numerator and denominator.
For instance, if we have a rational expression like \(\frac{x-3}{x^2-9}\), we want to simplify it as much as possible. Simplification often involves factoring polynomials in the numerator and denominator.
Difference of Squares
The difference of squares is a specific type of polynomial which follows the formula: \[a^2 - b^2 = (a - b)(a + b)\]. It is peculiar because it can always be factored into the product of two binomials.
In the exercise, we have the polynomial \(\frac{x-3}{x^2-9}\). The denominator \(x^2 - 9\) is a difference of squares. Thus, we can factor it as \(x^2 - 9 = (x - 3)(x + 3)\). Understanding this technique helps to simplify the rational expression to its core components.
In the exercise, we have the polynomial \(\frac{x-3}{x^2-9}\). The denominator \(x^2 - 9\) is a difference of squares. Thus, we can factor it as \(x^2 - 9 = (x - 3)(x + 3)\). Understanding this technique helps to simplify the rational expression to its core components.
- This can be especially handy when looking to simplify rational expressions
- or to solve equations involving polynomials.
Canceling Common Factors
Once we have factored the polynomials, the next step is to cancel out any common factors. For example, in our problem, once we factored the denominator, we had: \(\frac{x-3}{(x-3)(x+3)}\). Here, \(x-3\) appears in both the numerator and denominator.
Canceling common factors is a way to simplify the rational expression. After canceling \(x-3\), it simplifies to \(\frac{1}{x+3}\).
Remember, you can only cancel terms/factors, not individual terms. This step is essential to make any further solving or simplifying possible. It ensures the expression is in its simplest and most easily understandable form.
Canceling common factors is a way to simplify the rational expression. After canceling \(x-3\), it simplifies to \(\frac{1}{x+3}\).
Remember, you can only cancel terms/factors, not individual terms. This step is essential to make any further solving or simplifying possible. It ensures the expression is in its simplest and most easily understandable form.
Other exercises in this chapter
Problem 90
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View solution