Problem 53 - Introductory Algebra for College Students

Question

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

Step-by-Step Solution

Answer

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

Step 1: Rewrite the division as multiplication

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

Step 2: 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 \[ \frac{y(y^{2}+1)}{y(y-1)} \cdot \frac{(y-1)^{2}}{y^{2}(y-1)}\]

Step 3: 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 \[ \frac{y^{2}+1}{y} \cdot \frac{y-1}{y}\]

Step 4: Multiply the simplified rational expressions

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