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Chapter 7: Problem 53

Divide as indicated. $$\frac{y^{3}+y}{y^{2}-y} \div \frac{y^{3}-y^{2}}{y^{2}-2 y+1}$$

Short Answer

Expert verified
The simplified form of the expression is \((y^{2}+1)(y-1)\).

Step by step solution

01

Rewrite the division as multiplication

We change the division into a multiplication by taking the reciprocal of the divisor. Hence, \[ \frac{y^{3}+y}{y^{2}-y} \div \frac{y^{3}-y^{2}}{y^{2}-2 y+1} = \frac{y^{3}+y}{y^{2}-y} \cdot \frac{y^{2}-2 y+1}{y^{3}-y^{2}}\]
02

Factorize the polynomials

Now, we factorize the polynomials to make the simplification process easier. We obtain \[ y^{3}+y = y(y^{2}+1), \quad y^{2}-y = y(y-1), \quad y^{3}-y^{2} = y^{2}(y-1), \quad y^{2}-2 y+1 = (y-1)^{2}\] After substituting the factorized form we get \[ \frac{y(y^{2}+1)}{y(y-1)} \cdot \frac{(y-1)^{2}}{y^{2}(y-1)}\]
03

Simplify the rational expressions

We know that within fractions, expressions being multiplied can cancel out so we simplify the expression by cancelling out the common factors: \(y\), \(y-1\) in both the numerator and denominator. This results in \[ \frac{y^{2}+1}{y} \cdot \frac{y-1}{y}\]
04

Multiply the simplified rational expressions

Now multiply the fractions to obtain the final answer. Hence, the final result is \[ (y^{2}+1)(y-1)\]

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Most popular questions from this chapter

Factor: \(25 x^{2}-81 .\) (Section 6.4, Example 1)

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{4}{x-3}+\frac{2}{3-x}$$

perform the indicated operation. Where possible, reduce the answer to its lowest terms. $$\frac{1}{8}-\frac{5}{6}$$

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{x^{2}}{x-2}+\frac{4}{2-x}$$

Anthropologists and forensic scientists classify skulls using $$ \frac{L+60 W}{L}-\frac{L-40 W}{L} $$ where \(L\) is the skull's length and \(W\) is its width. a. Express the classification as a single rational expression. b. If the value of the rational expression in part (a) is less than \(75,\) a skull is classified as long. A medium skull has a value between 75 and \(80,\) and a round skull has a value over \(80 .\) Use your rational expression from part (a) to classify a skull that is 5 inches wide and 6 inches long.
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