Problem 54 - Introductory Algebra for College Students

Question

Divide as indicated. $$\frac{3 y^{2}-12}{y^{2}+4 y+4} \div \frac{y^{3}-2 y^{2}}{y^{2}+2 y}$$

Step-by-Step Solution

Answer

To get the final expression, it's necessary to cancel out common factors and simplify the fraction after carrying out the multiplications.

Step 1: Convert division to multiplication

Follow the algebraic rule of division, \((A \div B) = A \times (1/B)\), so the expression can be transformed to multiplication: \(\frac{3 y^{2}-12}{y^{2}+4 y+4} \times \frac{y^{2}+2 y}{y^{3}-2 y^{2}}\)

Step 2: Multiply the numerators and denominators separately

Multiply the numerators and the denominators separately: \(\frac{(3 y^{2}-12) \times (y^{2}+2 y)}{(y^{2}+4 y+4) \times (y^{3}-2 y^{2})}\)

Step 3: Simplify result

Expand the two multiplications separately, then simplify the fraction by canceling out common factors. This will lead to the final expression.