Problem 56
Question
Convert each rational expression into an equivalent rational expression that has the indicated denominator. $$\frac{x}{x-3}, \frac{?}{x^{2}-9}$$
Step-by-Step Solution
Verified Answer
\frac{x^2 + 3x}{x^2 - 9}
1Step 1: Factor the Denominator
First, factor the indicated denominator. The given denominator is \(x^2 - 9\). This is a difference of squares, which factors into \( (x - 3)(x + 3) \).
2Step 2: Adjust the Original Rational Expression
Next, adjust the original rational expression so it has the factored form \( (x - 3)(x + 3) \) as the new denominator. The original rational expression is \(\frac{x}{x-3}\). To achieve the new denominator, multiply both the numerator and the denominator by \( (x + 3) \).
3Step 3: Perform the Multiplication
Multiply the numerator and the denominator of \(\frac{x}{x-3}\) by \( (x + 3) \): \(\frac{x \times (x + 3)}{(x - 3) \times (x + 3)} = \frac{x(x + 3)}{x^2 - 9} \). This matches the indicated denominator \( x^2 - 9 \).
4Step 4: Simplify the Numerator
Simplify the expression in the numerator (if necessary). Here, it becomes \( x(x + 3) = x^2 + 3x \).
Key Concepts
Factoring PolynomialsRational ExpressionsDenominator Adjustment
Factoring Polynomials
Factoring polynomials is an essential skill in algebra. It involves rewriting a polynomial as a product of simpler polynomials. This is especially helpful when dealing with rational expressions.
In our exercise, we encounter the polynomial denominator \(x^2 - 9\). This specific form is known as a difference of squares. The general formula for factoring a difference of squares is:
\[ a^2 - b^2 = (a - b)(a + b) \]
Using this formula, we can factor \(x^2 - 9\) into:
\[ x^2 - 9 = (x - 3)(x + 3) \]
Factoring simplifies more complex operations, making it easier to manage and understand.
In our exercise, we encounter the polynomial denominator \(x^2 - 9\). This specific form is known as a difference of squares. The general formula for factoring a difference of squares is:
\[ a^2 - b^2 = (a - b)(a + b) \]
Using this formula, we can factor \(x^2 - 9\) into:
\[ x^2 - 9 = (x - 3)(x + 3) \]
Factoring simplifies more complex operations, making it easier to manage and understand.
Rational Expressions
Rational expressions are fractions where the numerator and the denominator are polynomials. Simplifying and manipulating these expressions is a key aspect of algebra.
Our initial rational expression is:
\[ \frac{x}{x - 3} \]
To convert this into an equivalent expression with a different denominator, we need to understand how to adjust both the numerator and denominator properly.
Key points to remember:
Our initial rational expression is:
\[ \frac{x}{x - 3} \]
To convert this into an equivalent expression with a different denominator, we need to understand how to adjust both the numerator and denominator properly.
Key points to remember:
- The numerator and denominator must be multiplied by the same term to ensure the value of the expression remains unchanged.
- Equivalent expressions have the same value, even though they may look different.
Denominator Adjustment
Denominator adjustment involves modifying the given rational expression to have the indicated new denominator.
In our example, the target denominator after factoring is \( (x - 3)(x + 3) \). To achieve this, the original expression \( \frac{x}{x-3} \) needs to be adjusted:
First, identify the missing factor in the original denominator that would result in the new denominator:
\( (x - 3)(x + 3) = x^2 - 9 \)
Since the original denominator is \( x - 3 \), we need to multiply both the numerator and denominator by \( x + 3 \):
\[ \frac{x}{x - 3} \cdot \frac{x + 3}{x + 3} = \frac{x(x + 3)}{(x - 3)(x + 3)} \]
This results in a new expression:
\[ \frac{x(x + 3)}{x^2 - 9} \]
Now, the numerator can be simplified if needed, ensuring the new expression matches the indicated denominator.
The key takeaway here is to always multiply both the numerator and denominator by the same term to keep the expression equivalent.
In our example, the target denominator after factoring is \( (x - 3)(x + 3) \). To achieve this, the original expression \( \frac{x}{x-3} \) needs to be adjusted:
First, identify the missing factor in the original denominator that would result in the new denominator:
\( (x - 3)(x + 3) = x^2 - 9 \)
Since the original denominator is \( x - 3 \), we need to multiply both the numerator and denominator by \( x + 3 \):
\[ \frac{x}{x - 3} \cdot \frac{x + 3}{x + 3} = \frac{x(x + 3)}{(x - 3)(x + 3)} \]
This results in a new expression:
\[ \frac{x(x + 3)}{x^2 - 9} \]
Now, the numerator can be simplified if needed, ensuring the new expression matches the indicated denominator.
The key takeaway here is to always multiply both the numerator and denominator by the same term to keep the expression equivalent.
Other exercises in this chapter
Problem 56
Solve each equation. $$\frac{2 m}{5}=\frac{10}{m}$$
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Perform the indicated operations. When possible write down only the answer. $$\left(x^{2}+y^{2}\right) \div(-1)$$
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Solve each equation. $$\frac{5}{4 x-2}-\frac{1}{1-2 x}=\frac{7}{3 x+6}$$
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Perform the indicated operations. When possible write down only the answer. $$\frac{x-y}{3} \cdot \frac{6}{y-x}$$
View solution