Problem 34
Question
Simplify each complex fraction. $$\frac{\frac{x+4}{x+1}+\frac{4}{x}}{\frac{x+1}{x}-\frac{1}{x+1}}$$
Step-by-Step Solution
Verified Answer
\( \frac{x^2 + 8x + 4}{x^2 + x + 1} \)
1Step 1: Simplify the numerator
Simplify the numerator \ \ \ \ Combine \( \frac{x+4}{x+1} + \frac{4}{x} \). Find a common denominator, which is \( x(x+1) \): \ \ \ \[ \frac{(x+4)x + 4(x+1)}{x(x+1)} = \frac{x^2+4x+4x+4}{x(x+1)} = \frac{x^2 + 8x + 4}{x(x+1)}. \ \ \] \
2Step 2: Simplify the denominator
Simplify the denominator \ \ Combine \( \frac{x+1}{x} - \frac{1}{x+1} \). Find a common denominator, which is \( x(x+1) \): \ \ \ \[ \frac{(x+1)(x+1) - x}{x(x+1)} = \frac{x^2 + 2x + 1 - x}{x(x+1)} = \frac{x^2 + x + 1}{x(x+1)}. \ \ \] \
3Step 3: Combine the simplified numerator and denominator
Combine the simplified numerator and denominator by dividing them: \ \[ \frac{ \frac{x^2 + 8x + 4}{x(x+1)} }{ \frac{x^2 + x + 1}{x(x+1)} } = \frac{x^2 + 8x + 4}{x^2 + x + 1} \ \ \] \ This is the final simplified form.
Key Concepts
common denominatorpolynomial divisionalgebraic expressions
common denominator
To simplify complex fractions, finding a common denominator is crucial. A common denominator allows you to combine fractional expressions by promoting them to have the same base.
For instance, consider adding \( \frac{x+4}{x+1} \) and \( \frac{4}{x} \). Both fractions have different denominators, \( x+1 \) and \( x \), respectively. To add them, you need the same denominator.
So, we look for the least common multiple of \( x \) and \( x+1 \). That’s \( x(x+1) \). Now, convert each fraction:
\( \frac{x+4}{x+1} \) becomes \( \frac{(x+4)x}{x(x+1)} \)
\( \frac{4}{x} \) becomes \( \frac{4(x+1)}{x(x+1)} \).
By changing the denominators to be the same, we can now combine these fractions:
\( \frac{(x+4)x + 4(x+1)}{x(x+1)} \).
After simplifying the numerator, you get \( \frac{x^2 + 8x + 4}{x(x+1)} \).
For instance, consider adding \( \frac{x+4}{x+1} \) and \( \frac{4}{x} \). Both fractions have different denominators, \( x+1 \) and \( x \), respectively. To add them, you need the same denominator.
So, we look for the least common multiple of \( x \) and \( x+1 \). That’s \( x(x+1) \). Now, convert each fraction:
\( \frac{x+4}{x+1} \) becomes \( \frac{(x+4)x}{x(x+1)} \)
\( \frac{4}{x} \) becomes \( \frac{4(x+1)}{x(x+1)} \).
By changing the denominators to be the same, we can now combine these fractions:
\( \frac{(x+4)x + 4(x+1)}{x(x+1)} \).
After simplifying the numerator, you get \( \frac{x^2 + 8x + 4}{x(x+1)} \).
polynomial division
Polynomial division is essential when simplifying fractions that include polynomial expressions. After combining fractions using a common denominator, you might need to divide polynomials to simplify further.
Consider you have \( \frac{ \frac{x^2 + 8x + 4}{x(x+1)} }{ \frac{x^2 + x + 1}{x(x+1)} } \). Here, both the numerator and the denominator are fractions.
To proceed, divide the numerator by the denominator:
\( \frac{x^2 + 8x + 4}{x(x+1)} \) divided by \( \frac{x^2 + x + 1}{x(x+1)} \)
Since both fractions share the common denominator \( x(x+1) \), these terms cancel each other out:
\( \frac{x^2 + 8x + 4}{x^2 + x + 1} \).
Thus, you’re left with the simplified polynomial fraction in its cleanest form.
Consider you have \( \frac{ \frac{x^2 + 8x + 4}{x(x+1)} }{ \frac{x^2 + x + 1}{x(x+1)} } \). Here, both the numerator and the denominator are fractions.
To proceed, divide the numerator by the denominator:
\( \frac{x^2 + 8x + 4}{x(x+1)} \) divided by \( \frac{x^2 + x + 1}{x(x+1)} \)
Since both fractions share the common denominator \( x(x+1) \), these terms cancel each other out:
\( \frac{x^2 + 8x + 4}{x^2 + x + 1} \).
Thus, you’re left with the simplified polynomial fraction in its cleanest form.
algebraic expressions
Understanding algebraic expressions will significantly aid in simplifying complex fractions. Algebraic expressions involve terms combined using operations like addition, subtraction, multiplication, and division.
When simplifying fractions like \( \frac{(x+4)x+4(x+1)}{x(x+1)} \), remember that each term in the numerator and denominator can be simplified individually. Breakdown the terms:
\( (x+4)x \) becomes \( x^2 + 4x \),
and \( 4(x+1) \) becomes \( 4x + 4 \).
Combining these simplifies the total expression in the numerator to:
\( x^2 + 4x + 4x + 4 \).
Similarly, handle subtraction in expressions such as \( \frac{x+1}{x} - \frac{1}{x+1} \).
After finding a common denominator, \( x(x+1) \), combine terms by expanding and subtracting:
\( (x+1)(x+1) \) becomes \( x^2 + 2x + 1 \),
then subtract \( x \).
This results in \( x^2 + x + 1 \). Simplifying algebraic expressions step-by-step ensures clarity and accuracy.
When simplifying fractions like \( \frac{(x+4)x+4(x+1)}{x(x+1)} \), remember that each term in the numerator and denominator can be simplified individually. Breakdown the terms:
\( (x+4)x \) becomes \( x^2 + 4x \),
and \( 4(x+1) \) becomes \( 4x + 4 \).
Combining these simplifies the total expression in the numerator to:
\( x^2 + 4x + 4x + 4 \).
Similarly, handle subtraction in expressions such as \( \frac{x+1}{x} - \frac{1}{x+1} \).
After finding a common denominator, \( x(x+1) \), combine terms by expanding and subtracting:
\( (x+1)(x+1) \) becomes \( x^2 + 2x + 1 \),
then subtract \( x \).
This results in \( x^2 + x + 1 \). Simplifying algebraic expressions step-by-step ensures clarity and accuracy.
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