Question

Multiply or divide the following rational expressions, as indicated, and simplify your answer:

The expression (y² - 4y + 3) / (y² + 3y - 18) is divided by (y + 8) / (y + 8). Which of the following represents the simplified result?

a) (y - 1) / (y + 6)

b) (y + 6) / (y - 1)

c) (y + 1) / (y - 6)

d) (y - 6) / (y + 1)

Answer 1

The simplified result of the given expression is (y - 1) / (y + 6). We factored both the numerator and denominator and then cancelled out the common factors. The final expression is option (a).

Step-by-step explanation:

To find the simplified result of the given expression, which involves dividing rational expressions, we need to perform the division operation. Since division by a fraction is the same as multiplication by its reciprocal, we'll multiply the first expression by the reciprocal of the second. The given expression is (y² - 4y + 3) / (y² + 3y - 18) divided by (y + 8) / (y + 8).

First, notice that the second expression (y + 8) / (y + 8) simplifies to 1, since the numerator and denominator are the same. So, the division of the whole expression by (y + 8) / (y + 8) does not change the expression. Hence, we only need to consider the first expression.

Let's factorize the numerator and denominator of the first expression:

• The numerator y² - 4y + 3 can be factorized into (y - 1)(y - 3).
• The denominator y² + 3y - 18 can be factorized into (y - 3)(y + 6).

After factorization, our expression looks like:

(y - 1)(y - 3) / (y - 3)(y + 6)

Canceling out the common factors (y - 3) from the numerator and denominator, we get:

(y - 1) / (y + 6)

The simplified result is, therefore, option (a).