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Chapter 8: Problem 115

\(\text { Divide: } \frac{1}{x^{2}-17 x+30} \div \frac{1}{x^{2}+7 x-18}\)

Short Answer

Expert verified
The simplified form of the division of the two given fractions is \( \frac{x+9}{x-15}\)

Step by step solution

01

Write the division as a multiplication

To start, convert the division into a multiplication problem by flipping, or taking the reciprocal of, the fraction we are dividing by. This results in \(\frac{1}{x^{2}-17 x+30} * \frac{x^{2}+7 x-18}{1}\)
02

Simplify the multiplication

This multiplication results in the expression \(\frac{x^{2}+7 x-18}{x^{2}-17 x+30}\)
03

Factor the polynomials

Factorizing the expressions in the numerator and denominator, you get \(\frac{(x-2)(x+9)}{(x-15)(x-2)}\)
04

Cancel out common factors

Factor out common terms in the numerator and the denominator. This leaves \( \frac{x+9}{x-15}\) only as the (x-2) in the numerator and denominator cancel each other out

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Key Concepts

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

Polynomial Division
Polynomial division can initially seem challenging, but breaking it down step by step makes it approachable. When you encounter an algebraic fraction that involves division, the first step is to turn the problem into a multiplication task. This is done by using the reciprocal of the divisor (the second fraction).

Remember, dividing by a fraction is the same as multiplying by its reciprocal. For example, dividing by \( \frac{a}{b} \) is equivalent to multiplying by \( \frac{b}{a} \). This crucial step allows us to use the rules of multiplication to continue solving the problem. Here鈥檚 a quick recap of the process:
  • Identify the divisor (the fraction you're dividing by).
  • Find the reciprocal of the divisor (flip the fraction).
  • Change the operation from division to multiplication using the reciprocal.
Factoring Polynomials
Factoring polynomials is a fundamental skill in algebra, helping simplify expressions and solutions. Factoring transforms a complex expression into its simpler components or products, revealing common factors or zeros.

To factor a quadratic polynomial like those in our exercise's numerator and denominator, look for two numbers that multiply to the constant term and add to the middle term (the coefficient of \( x \)). For example:
  • Consider \( x^2 - 17x + 30 \). We need two numbers that multiply to 30 and add to -17; these numbers are -15 and -2. Hence, we factor it as \( (x-15)(x-2) \).
  • Similarly, for \( x^2 + 7x - 18 \), look for numbers that multiply to -18 and add to 7, which are 9 and -2. Thus, it factors as \( (x-2)(x+9) \).
Recognizing patterns and practicing will improve your factoring skills, leading to quicker problem-solving.
Simplifying Fractions
Simplifying fractions is about reducing them to their simplest form, making them easier to interpret and use. This involves canceling out any common factors shared by the numerator and the denominator.

After factoring the polynomials, we notice that both the numerator and denominator include \( (x-2) \) as a factor. By cancelling these common terms, we streamline the fraction. Here is the simplified process:
  • Identify common factors in the numerator and denominator. In this case, \( (x-2) \) appears in both.
  • Cancel out these common factors, simplifying the expression. This step gives us the reduced expression \( \frac{x+9}{x-15} \).
To ensure accuracy, particularly in exams or assignments, check your work, as this process needs careful handling to avoid missteps.

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Most popular questions from this chapter

In Exercises \(53-74\), rationalize each denominator. Simplify, if possible. $$\frac{1}{5+\sqrt{2}}$$

In Exercises \(75-82\), rationalize each denominator. Simplify, if possible $$\frac{2 x+4-2 h}{\sqrt{x+2-h}}$$

When a radical expression has its denominator rationalized, we change the denominator so that it no longer contains a radical. Doesn't this change the value of the radical expression? Explain.

Simplify by first writing the expression in radical form. If applicable, use a calculator to verify your answer. $$81^{-\frac{5}{4}}$$

In Exercises \(75-82\), rationalize each denominator. Simplify, if possible $$\frac{\sqrt{2}}{\sqrt{7}}+\frac{\sqrt{7}}{\sqrt{2}}$$
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