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Chapter 0: Problem 32

Multiply or divide as indicated. $$\frac{x^{3}-25 x}{4 x^{2}} \cdot \frac{2 x^{2}-2}{x^{2}-6 x+5} \div \frac{x^{2}+5 x}{7 x+7}$$

Short Answer

Expert verified
So, the result of the operation is \(-\frac{(x^{2}-25)(x^{2}-1)(7)(x+1)}{x^{3}-\frac{5}{2}x}\)

Step by step solution

01

Simplify the Fractions

As seen in the analysis, observe common factors and simplify each fraction before performing any operation:1) For the first fraction \(\frac{x^{3}-25 x}{4 x^{2}}\), factor out \(x\) in the numerator: \(\frac{x(x^{2}-25)}{4 x^{2}}\).2) For the second fraction \(\frac{2 x^{2}-2}{x^{2}-6 x+5}\), factor out 2 in the numerator: \(\frac{2(x^{2}-1)}{x^{2}-6x+5}\).3) For the third fraction, \(\frac{x^{2}+5 x}{7x+7}\), factor out \(x\) in the numerator and 7 in the denominator: \(\frac{x(x+5)}{7(x+1)}\).
02

Multiply and Divide Fractions

Now multiply the first two fractions and divide by the third:\(\frac{x(x^{2}-25)}{4 x^{2}} \cdot \frac{2(x^{2}-1)}{x^{2}-6x+5} \div \frac{x(x+5)}{7(x+1)}\)Perform the multiplication:\(\frac{x(x^{2}-25) \cdot 2(x^{2}-1)}{4 x^{2} \cdot (x^{2}-6x+5)}\)And then perform the division:\(\frac{x(x^{2}-25) \cdot 2(x^{2}-1)}{4 x^{2} \cdot (x^{2}-6x+5)} \cdot \frac{7(x+1)}{x(x+5)}\)
03

Simplify the Expression

Observing the expression, notice that there is an \(x\) in the numerator and denominator that can be cancelled out. Also split the denominator to cancel other terms:\(\frac{2x(x^{2}-25)(x^{2}-1)(7)(x+1)}{4x[x^{2}(x+5)+(x^{2}-6x+5)]}\)Simplify:\(\frac{2(x^{2}-25)(x^{2}-1)(7)(x+1)}{4x^{3}+5x^{2}-6x^{2}+5x}\)This simplifies to:\(\frac{2(x^{2}-25)(x^{2}-1)(7)(x+1)}{-2x^{3}+5x}\)Finally, reduce the expression by dividing the numerator and denominator by -2:\(\frac{-(x^{2}-25)(x^{2}-1)(7)(x+1)}{x^{3}-\frac{5}{2}x}\)

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