Problem 63 - Introductory Algebra for College Students

Question

Divide as indicated. $$\frac{x y-y^{2}}{x^{2}+2 x+1} \div \frac{2 x^{2}+x y-3 y^{2}}{2 x^{2}+5 x y+3 y^{2}}$$

Step-by-Step Solution

Answer

\(\frac{(x y-y^{2}) \cdot (2 x^{2}+5 x y+3 y^{2})}{(x^{2}+2 x+1) \cdot (2 x^{2}+x y-3 y^{2})}\)

Step 1: Reinterpret the division as a multiplication

Rewrite the problem as a multiplication: \(\frac{x y-y^{2}}{x^{2}+2 x+1} \cdot \frac{2 x^{2}+5 x y+3 y^{2}}{2 x^{2}+x y-3 y^{2}}\)

Step 2: Multiply the numerators

Multiply the numerators of both fractions together: \( (x y-y^{2}) \cdot (2 x^{2}+5 x y+3 y^{2}) \)

Step 3: Multiply the denominators

Multiply the denominators of both fractions together: \( (x^{2}+2 x+1) \cdot (2 x^{2}+x y-3 y^{2}) \)

Step 4: Simplify if possible

Simplify the result if possible. In this case, no further simplification can be done.