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Chapter 5: Problem 7

Simplify \(\frac{x+2}{x^{2}+9 x+20} \times \frac{x+5}{x+2}\).

Short Answer

Expert verified
Question: Simplify the given product of rational expressions: \(\frac{x+2}{x^2 + 9x + 20} \times \frac{x+5}{x+2}\) Answer: \(\frac{1}{x+4}\)

Step by step solution

01

Factor the denominators

First, we need to factor the quadratic expressions in the denominators. Observe that the first one can be factored into \((x+4)(x+5)\) and the second one is just a linear expression. So the given expression can be written as: $$\frac{x+2}{(x+4)(x+5)} \times \frac{x+5}{x+2}$$
02

Cancel common factors

We can see that both the numerator and the denominator have common factors, and they can be canceled out. We can cancel \((x+2)\) and \((x+5)\) from the numerator and the denominator, respectively. $$\frac{\cancel{(x+2)}}{(x+4)(\cancel{(x+5)})} \times \frac{\cancel{(x+5)}}{\cancel{(x+2)}}$$
03

Simplify

After canceling the common factors, we are left with the following simplified expression: $$\frac{1}{x+4}$$ So, the simplified form of the given expression is: \(\frac{1}{x+4}\).

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Most popular questions from this chapter

Express \(\frac{1}{s}+\frac{1}{s^{2}}\) as a single fraction.

Remove the brackets from the given expression: \((x-1)(x+2)\)

For the following formulae, find \(y\) at the given values of \(x\) : (a) \(y=\frac{1}{2} x+\frac{1}{3}, x=-2, x=0, x=1\) (b) \(y=2 x^{2}+3 x+1, x=-2, x=-1\), \(x=0, x=1, x=2\) (c) \(y=2 x^{3}+3 x^{2}, x=-4, x=0, x=4\)

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