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Chapter 8: Problem 101

Find the quotient: \(\frac{x^{2}-6 x+9}{x^{2}-x-6} \div \frac{x^{2}+2 x-15}{x^{2}+2 x}\).

Short Answer

Expert verified
Question: Divide the given algebraic fractions and simplify the expression: \(\frac{x^{2}-6 x+9}{x^{2}-x-6} \div \frac{x^{2}+2x-15}{x^{2}+2 x}\). Answer: \(\frac{x(x-3)}{(x+2)(x+5)}.\)

Step by step solution

01

Simplify the given expression and rewrite division as multiplication

To simplify, factor the given expressions and rewrite as multiplication by taking the reciprocal of the second fraction. We have the given expression as: \(\frac{x^{2}-6 x+9}{x^{2}-x-6} \div \frac{x^{2}+2x-15}{x^{2}+2 x}\). Factor each expression: \(\frac{(x-3)^2}{(x-3)(x+2)} \div \frac{(x+5)(x-3)}{x(x+2)}.\) Now, rewrite the division as multiplication by finding the reciprocal of the second fraction: \(\frac{(x-3)^2}{(x-3)(x+2)} \times \frac{x(x+2)}{(x+5)(x-3)}.\)
02

Cancel out common factors and multiply

Now, cancel out the common factors in the numerator and the denominator. \(\frac{\cancel{(x-3)}(x-3)}{\cancel{(x-3)}(x+2)} \times \frac{x\cancel{(x+2)}}{(x+5)\cancel{(x-3)}}.\) After canceling out the common factors, we have: \(\frac{(x-3)}{(x+2)} \times \frac{x}{(x+5)}.\) Now, multiply the simplified fractions: \(\frac{(x-3) \times x}{(x+2) \times (x+5)}.\)
03

Write the final answer

After multiplying the fractions, the final answer is: \(\frac{x(x-3)}{(x+2)(x+5)}.\)

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Key Concepts

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

Factoring Polynomials
Polynomials are expressions that consist of variables and coefficients. Factoring them involves rewriting the polynomial as a product of simpler polynomials. When you factor a polynomial, you essentially break it down into its building blocks.For example, if you have a quadratic expression like \(x^2 - 6x + 9\), try to find two numbers that multiply to the last term (9) and add up to the middle term (-6). Here, these numbers are -3 and -3, giving us \((x-3)^2\) as the factored form.
  • Determine if the polynomial is a quadratic trinomial. Use trial and error or structured methods like the AC method or completing the square.
  • Remember the basic forms such as difference of squares: \(a^2 - b^2 = (a-b)(a+b)\).
  • Factor completely by checking for the greatest common factor (GCF) first.
Factoring is crucial in simplifying expressions and solving polynomial equations.
Rational Expressions
A rational expression is a fraction where both the numerator and the denominator are polynomials. Understanding how to manipulate these expressions is vital in algebra.When dealing with division of rational expressions, you can rewrite the division as multiplication by the reciprocal. In the given example, when dividing \(\frac{x^{2}-6 x+9}{x^{2}-x-6}\) by \(\frac{x^{2}+2x-15}{x^{2}+2 x}\), it transforms into multiplication:
  • Take the reciprocal of the second rational expression to turn division into multiplication.
  • Ensure all expressions are factored completely before proceeding with any operations.
  • Polynomials in the numerator and denominator can be complex, so pay close attention to factoring correctly.
Manipulating rational expressions often involves applying this principle of turning division into multiplication, which simplifies the process considerably.
Simplifying Expressions
Simplifying expressions is about making them easier to understand and work with. For rational expressions, this often involves canceling out common factors in the numerator and the denominator.Once you have both rational expressions prepared as multiplication, check for common factors:
  • Identify terms that appear in both the numerator and the denominator, such as \(x-3\) in this exercise.
  • Cancel these common factors, which simplifies the expression. Remember, canceling only applies if the factor is a full term.
  • Multiply any remaining terms to get the final, simplified expression.
In our example, after canceling \(x-3\) and \(x+2\), you're left with a simpler form that reflects the essence of the original problem without excess factors.

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Most popular questions from this chapter

For the following problems, perform the divisions. $$ \frac{3 a^{2}+4 a-4}{a^{2}+3 a+3} $$

For the following problems, divide the polynomials. $$ -4 b^{7}-3 b^{6}-22 b^{5}-19 b^{4}+12 b^{3}-6 b^{2}+b+4 \text { by } b^{2}+6 $$

For the following problems, perform the divisions. $$ \left(2 x^{5}+5 x^{4}-1\right) \div(2 x+5) $$

For the following problems, perform the indicated operations. $$ \frac{x^{2}-x-12}{x^{2}-3 x+2} \cdot \frac{x^{2}+3 x-4}{x^{2}-3 x-18} $$

For the following problems, perform the indicated operations. $$ \frac{y^{2}-1}{y^{2}+9 y+20} \div \frac{y^{2}+5 y-6}{y^{2}-16} $$
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