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Chapter 0: Problem 30

Multiply or divide as indicated. $$\frac{x^{2}-4}{x^{2}+3 x-10} \div \frac{x^{2}+5 x+6}{x^{2}+8 x+15}$$

Short Answer

Expert verified
The result of the complex fractions operation is 1.

Step by step solution

01

Factorize the Expressions

Firstly, factor all the expressions in the operations: \(x^{2}-4\) can be rewritten as \((x-2)(x+2)\), \(x^{2}+3x-10\) can be rewritten as \((x-2)(x+5)\), \(x^{2}+5x+6\) can be rewritten as \((x+2)(x+3)\), and \(x^{2}+8x+15\) can be rewritten as \((x+3)(x+5)\).
02

Rewrite the Division Operation

Substitute the factored expressions back into the original operation and transform the division operation into a multiplication one by swapping the second fraction (multiply by reciprocal):\(\frac{(x-2)(x+2)}{(x-2)(x+5)} \times \frac{(x+3)(x+5)}{(x+2)(x+3)} \)
03

Simplify the Expression

Finally, cancel the common factors from the numerator and denominator of the resulting expression: \( \frac{(x-2)(x+2)}{(x-2)(x+5)} \times \frac{(x+3)(x+5)}{(x+2)(x+3)} = 1 \)
04

Conclusion

The final result of the division operation after factoring and canceling out the common elements is 1.

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Most popular questions from this chapter

Will help you prepare for the material covered in the first section of the next chapter. If \(y=4-x,\) find the value of \(y\) that corresponds to values of \(x\) for each integer starting with -3 and ending with 3

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What is the discriminant and what information does it provide about a quadratic equation?

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\frac{2 x-1}{x-7}+\frac{3 x-1}{x-7}-\frac{5 x-2}{x-7}=0$$

Explain how to simplify a rational expression.
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