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Magazine Chapter 4: Problem 20
Perform the indicated operations, and express your answers in simplest form. $$ \frac{7}{x^{2}+1}-\frac{3}{x^{2}+7 x-60} $$
Short Answer
Expert verified
The simplest form is \( \frac{4x^2 + 49x - 423}{(x^2 + 1)(x + 12)(x - 5)} \).
Step by step solution
01
Factor the Quadratic Expression
The denominator \( x^2 + 7x - 60 \) can be factored by finding two numbers that multiply to -60 and add to 7. These numbers are 12 and -5. Therefore, \( x^2 + 7x - 60 \) factors into \( (x + 12)(x - 5) \).
02
Find the Least Common Denominator (LCD)
The least common denominator of the fractions \( \frac{7}{x^2+1} \) and \( \frac{3}{(x+12)(x-5)} \) is the product of both denominators, since they have no common factors. Thus, the LCD is \( (x^2 + 1)(x + 12)(x - 5) \).
03
Rewrite Each Fraction with the LCD
Convert each fraction to have the LCD as its denominator:- For \( \frac{7}{x^2+1} \), multiply the numerator and the denominator by \( (x + 12)(x - 5) \), giving \( \frac{7(x + 12)(x - 5)}{(x^2 + 1)(x + 12)(x - 5)} \).- For \( \frac{3}{(x + 12)(x - 5)} \), multiply the numerator and the denominator by \( x^2 + 1 \), producing \( \frac{3(x^2 + 1)}{(x^2 + 1)(x + 12)(x - 5)} \).
04
Combine the Fractions
Now that both fractions have the same denominator, we can subtract the numerators:\[ \frac{7(x + 12)(x - 5) - 3(x^2 + 1)}{(x^2 + 1)(x + 12)(x - 5)} \].
05
Simplify the Numerator
Expand and simplify the numerator:- Expand \( 7(x + 12)(x - 5) \) to get \( 7(x^2 + 7x - 60) = 7x^2 + 49x - 420 \).- Expand \( 3(x^2 + 1) \) to get \( 3x^2 + 3 \).Subtracting these gives:\[ (7x^2 + 49x - 420) - (3x^2 + 3) = 4x^2 + 49x - 423 \].
06
Finalize the Expression
The simplified expression of the original problem is:\[ \frac{4x^2 + 49x - 423}{(x^2 + 1)(x + 12)(x - 5)} \].Since the numerator \( 4x^2 + 49x - 423 \) does not factor further in such a way that it would have a common factor with the denominator, this is the simplest form.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Factoring Quadratics
Factoring quadratic expressions is like solving a puzzle using multiplication. When you see a quadratic like \( x^2 + 7x - 60 \), your goal is to find two numbers that multiply to the constant term, which in this case is -60, and simultaneously add up to the coefficient of the linear term, which is 7.
Here's a simple way to approach this:
Here's a simple way to approach this:
- List the factor pairs of -60. These include: (-1, 60), (1, -60), (-2, 30), (2, -30), (-3, 20), (3, -20), (-4, 15), (4, -15), (-5, 12), and (5, -12).
- Look for a pair that sums to 7. If we check, we see 12 and -5 multiply to -60 but add up to 7.
Common Denominator
A common denominator is the key to combining fractions. Think of it like choosing a common unit to compare two different quantities.
For algebraic fractions like \( \frac{7}{x^2+1} \) and \( \frac{3}{(x+12)(x-5)} \), finding the least common denominator (LCD) helps in lining up denominators for easier operations:
For algebraic fractions like \( \frac{7}{x^2+1} \) and \( \frac{3}{(x+12)(x-5)} \), finding the least common denominator (LCD) helps in lining up denominators for easier operations:
- First, identify the unique factors in each denominator. For \( x^2 + 1 \), there are no factorizable components. However, \( x^2 + 7x - 60 \) is already factored into \( (x+12)(x-5) \).
- Multiply together all unique denominators. Since \( x^2+1 \) and \( (x+12)(x-5) \) have no common factors, the LCD is simply \( (x^2+1)(x+12)(x-5) \).
Simplifying Expressions
Simplifying expressions is about making them as concise as possible while retaining their original value. After acquiring a common denominator, expressions can be combined through subtraction.
For the fractions \( \frac{7(x + 12)(x - 5)}{(x^2 + 1)(x + 12)(x - 5)} \) and \( \frac{3(x^2 + 1)}{(x^2 + 1)(x + 12)(x - 5)} \), the goal is to combine these into one fraction then simplify:
For the fractions \( \frac{7(x + 12)(x - 5)}{(x^2 + 1)(x + 12)(x - 5)} \) and \( \frac{3(x^2 + 1)}{(x^2 + 1)(x + 12)(x - 5)} \), the goal is to combine these into one fraction then simplify:
- Subtract the numerators directly: \( 7(x + 12)(x - 5) - 3(x^2 + 1) \).
- Use expansion to simplify: Result in \( 4x^2 + 49x - 423 \) after careful subtraction of terms.
