Problem 64 - Introductory Algebra for College Students

Question

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

Step-by-Step Solution

Answer

1

Step 1: Reformulate the division as multiplication

Change the division operation to multiplication by taking the reciprocal of the second fraction. The given exercise is equivalent to: \[\frac{x^{2}-4 y^{2}}{x^{2}+3 x y+2 y^{2}} * \frac{x+y}{x^{2}-4 x y+4 y^{2}}\]

Step 2: Factorize the polynomials

Factorize the polynomials in the numerators and denominators of both fractions.\[\frac{(x-2y)(x+2y)}{(x+y)(x+2y)} * \frac{x+y}{(x-2y)^2}\]

Step 3: Cancel out common terms

Cancel out the common factors in both the numerator and the denominator.\[\frac{x-2y}{x+2y} * \frac{x+y}{x-2y}\]

Step 4: Further simplifying

Again, cancel out common factors in both the numerator and the denominator\[\frac{x+2y}{x+2y}\]

Step 5: The final answer

The common terms further cancel out giving a scalar quantity. \[1\]