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Chapter 7: Problem 70

Add or subtract as indicated. Simplify the result, if possible. $$\frac{x^{2}+9 x}{x^{2}-2 x-3}+\frac{5}{3-x}$$

Short Answer

Expert verified
The simplified result of the addition is \((x^2 + 9x - 5) / (x-3)(x+1)\)

Step by step solution

01

Factorize the Denominators

The first step would be to factorize the denominator of each fraction, where possible. This helps to find a common denominator and simplify the problem. For \(x^{2}-2x-3\), factorization results in \((x-3)(x+1)\). The second fraction \(5/(3-x)\) can be written as \(-5/(x-3)\) to align with the first denominator.
02

Find a Common Denominator

The common denominator can be found by taking the Least Common Multiple (LCM) of the two denominators. The LCM of \((x-3)(x+1)\) and \((x-3)\) is \((x-3)(x+1)\).
03

Simplify the Numerators

With a common denominator, the next step would be to simplify the numerators by adding them together. This results in \((x^{2}+9x - 5)\) on the numerator and \((x-3)(x+1)\) on the denominator.
04

Simplification of the Fraction

Now that both fractions have a common denominator, simplify the expression by combining the numerators, leading to: \((x^2 + 9x - 5) / (x-3)(x+1)\).

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Key Concepts

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

Factorization
Factorization is a crucial step in simplifying rational expressions. It involves breaking down a polynomial into its component factors. To factorize the polynomial in the denominator of the first fraction, identify two numbers whose product equals the constant term and whose sum equals the coefficient of the middle term. In the expression \(x^2 - 2x - 3\), factor it to \((x - 3)(x + 1)\). By recognizing common patterns and products, solving becomes easier. Remember that simpler forms can later be used to find a common denominator.
Common Denominator
Finding a common denominator is essential when adding or subtracting rational expressions. It allows you to combine fractions into a single expression. A common denominator is the same as finding a shared base for both fractions. In our problem, the denominators are \((x-3)(x+1)\) and \((3-x)\).
  • Notice \((3-x)\) can be rewritten as \(-(x-3)\) to match the form \((x-3)\).
  • This trick uses the property that multiplication by -1 flips subtraction terms.
This simplification helps in aligning both denominators and preparing for the next steps.
Least Common Multiple
The Least Common Multiple (LCM) for expressions simplifies finding a common denominator. It is the smallest expression that both numbers can divide evenly. In terms of our rational expressions, consider:
  • First expression's denominator: \((x-3)(x+1)\).
  • Second expression's rewritten denominator: \((x-3)\).
Find the LCM by taking the unique factors from each, resulting in \((x-3)(x+1)\). Identifying the LCM ensures you do not miss any necessary factors in the denominators. This step is vital for keeping the process orderly and accurate.
Addition and Subtraction of Rational Expressions
Addition and subtraction of rational expressions require aligned denominators, now covered by our common denominator. Once achieved, the numerators can be simply added or subtracted. For our expressions
  • The first became: \((x^2+9x)/((x-3)(x+1))\)
  • The second became: \(-5/(x-3)\) rewritten to align with our denominator.
Combine the numerators:
  • Add \(x^2+9x\) and subtract \(-5\), yielding \(x^2+9x-5\).
Write the combination over the common denominator \((x-3)(x+1)\). This process of combining enables simplification, leading towards a solution.

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Most popular questions from this chapter

Add or subtract as indicated. Simplify the result, if possible. $$\frac{x+7}{4 x+12}+\frac{x}{9-x^{2}}$$

Add or subtract as indicated. Simplify the result, if possible. $$\frac{7}{5 y^{2}-5 y}-\frac{2}{5 y-5}$$

Add or subtract as indicated. Simplify the result, if possible. $$\frac{x-1}{x}+\frac{y+1}{y}$$

Will help you prepare for the material covered in the first section of the next chapter. Here are two sets of ordered pairs: $$\text { set } 1:\\{(1,5),(2,5)\\}$$ set \(2:[(5,1),(5,2)]\) In which set is each \(x\) -coordinate paired with only one \(y\) -coordinate?

Factor completely: \(x^{4}+2 x^{3}-3 x-6 .\) (Section 6.1 Example 8 )
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