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Chapter 6: Problem 108

Let \(P\), \(Q\), and \(R\) be rational expressions defined as follows. $$P=\frac{6}{x+3}, \quad Q=\frac{5}{x+1}, \quad R=\frac{4 x}{x^{2}+4 x+3}$$ Simplify the complex fraction \(\frac{P+Q}{R}\)

Short Answer

Expert verified
\( \frac{11}{4} + \frac{21}{4x} \)

Step by step solution

01

- Identify the expressions

Given rational expressions are:\[P=\frac{6}{x+3}, \ Q=\frac{5}{x+1}, \ R=\frac{4 x}{x^{2}+4 x+3}\]
02

- Simplify the denominator of R

Factor the denominator of expression R:\[x^{2}+4 x+3 = (x+3)(x+1)\] Thus, \[ R = \frac{4x}{(x+3)(x+1)}\]
03

- Write expression for \ \( \frac{P+Q}{R} \)

Now substitute the simplified values:\[ \frac{P+Q}{R} = \frac{\frac{6}{x+3} + \frac{5}{x+1}}{\frac{4x}{(x+3)(x+1)}} \]
04

- Find common denominator for P and Q

The common denominator for \(P\) and \(Q\) is \((x+3)(x+1)\):\[ \frac{6}{x+3} = \frac{6(x+1)}{(x+3)(x+1)} \] \[ \frac{5}{x+1} = \frac{5(x+3)}{(x+3)(x+1)} \] Therefore,\[ P + Q = \frac{6(x+1) + 5(x+3)}{(x+3)(x+1)} = \frac{6x+6 + 5x + 15}{(x+3)(x+1)} = \frac{11x+21}{(x+3)(x+1)} \]
05

- Simplify the fraction

Since both the numerator \(P + Q\) and the denominator \(R\) have the same denominator \((x+3)(x+1)\), the complex fraction simplifies as follows: \[ \frac{P+Q}{R} = \frac{\frac{11x+21}{(x+3)(x+1)}}{\frac{4x}{(x+3)(x+1)}} = \frac{11x+21}{4x} \]
06

- Simplify the result

Simplify the resulting fraction: \[ \frac{11x+21}{4x} = \frac{11x}{4x} + \frac{21}{4x} = \frac{11}{4} + \frac{21}{4x} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rational Expressions
A rational expression is a fraction where both the numerator and the denominator are polynomials. For example, in the exercise, the expressions given were:
  • P = \(\frac{6}{x+3}\)
  • Q = \(\frac{5}{x+1}\)
  • R = \(\frac{4 x}{x^{2}+4 x+3}\)
Rational expressions are mathematical tools you will often encounter. They follow the same rules as regular fractions. This means we can add, subtract, multiply, or divide them by finding common denominators and simplifying.
Factoring Polynomials
Factoring polynomials involves expressing a polynomial as a product of its linear factors. It helps in simplifying complex fractions. For instance, part of the solution requires simplifying the expression R. Given:
  • \(R = \frac{4 x}{x^{2}+4 x+3}\)
We factor the quadratic polynomial in the denominator:
  • \(x^{2}+4 x+3 = (x+3)(x+1)\)
Thus, simplifying R:
  • \(R = \frac{4 x}{(x+3)(x+1)}\)
Factoring is a crucial step since it allows us to identify common factors and simplify the entire expression effectively.
Common Denominators
Finding common denominators is essential when you add or subtract rational expressions. In our case, we needed to find a common denominator for P and Q before summing them. Given:
  • \(\frac{6}{x+3}\) and \(\frac{5}{x+1}\)
The common denominator is \((x+3)(x+1)\). We rewrite each fraction with this common denominator:
  • \(\frac{6}{x+3} = \frac{6(x+1)}{(x+3)(x+1)}\)
  • \(\frac{5}{x+1} = \frac{5(x+3)}{(x+3)(x+1)}\)
Adding these fractions then gives us:
  • \(\frac{6(x+1) + 5(x+3)}{(x+3)(x+1)} = \frac{11x+21}{(x+3)(x+1)}\)
Finding a common denominator helps combine these fractions into a single rational expression.
Simplification
Simplification involves reducing expressions to their simplest form. After obtaining \(\frac{P+Q}{R}\), the goal is to simplify:
  • \( \frac{P+Q}{R} = \frac{\frac{11x+21}{(x+3)(x+1)}}{\frac{4x}{(x+3)(x+1)}}\)
  • Since both the numerator and the denominator share the same denominator \((x+3)(x+1)\), it simplifies to \(\frac{11x+21}{4x}\).
Next, by breaking the fraction:
  • \(\frac{11x+21}{4x} = \frac{11x}{4x} + \frac{21}{4x}= \frac{11}{4}+\frac{21}{4x}\)
Simplification makes the expression easier to work with and understand. It involves combining like terms, reducing fractions, and eliminating common factors.

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Most popular questions from this chapter

Simplify each fraction. $$ \frac{1+t^{-1}-56 t^{-2}}{1-t^{-1}-72 t^{-2}} $$

Solve each formula or equation for the specified variable. $$ \frac{t}{x-1}-\frac{2}{x+1}=\frac{1}{x^{2}-1} \text { for } t $$

Find the slope of the line that passes through each pair of points. This will involve simplifying complex fractions. $$ \left(\frac{1}{2}, \frac{5}{12}\right) \text { and }\left(\frac{1}{4}, \frac{1}{3}\right) $$

Simplify each complex fraction. Use either method. $$ \frac{\frac{5}{x^{2} y}-\frac{2}{x y^{2}}}{\frac{3}{x^{2} y^{2}}+\frac{4}{x y}} $$

These exercises involve grouping symbols, factoring by grouping, and factoring sums and differences of cubes. Multiply or divide as indicated. Write each answer in lowest terms. \(\frac{m-8}{m-4} \div\left(\frac{m^{2}-12 m+32}{8 m} \cdot \frac{m^{2}-8 m}{m^{2}-8 m+16}\right)\)
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