Divide and Simplify
y^2-16/y divide y-4/y+1
To solve this expression, "Divide and Simplify" means we will perform the division and then simplify the resulting expression.
First, let's focus on the division part. To divide two fractions, we need to follow these steps:
Step 1: Flip the second fraction (also called the divisor), so it becomes the reciprocal. In this case, the second fraction is (y - 4) / (y + 1), and its reciprocal is (y + 1) / (y - 4).
Step 2: Multiply the first fraction (also called the dividend) by the reciprocal of the second fraction. So, we have:
(y^2 - 16) / y * (y + 1) / (y - 4)
Step 3: Simplify the resulting expression by canceling out any common factors present in the numerator and the denominator. Notice that (y - 4) is common to both the numerator and denominator, so we can cancel it out:
(y^2 - 16) / y * (y + 1) / (y - 4) = (y^2 - 16) / y * (y + 1) / (y - 4) = (y^2 - 16) * (y + 1) / (y * (y - 4))
Now, let's simplify the expression further:
The expression (y^2 - 16) represents a difference of squares, which can be factored as follows:
y^2 - 16 = (y - 4)(y + 4)
So, substituting this factorization into the equation:
(y - 4)(y + 4) * (y + 1) / (y * (y - 4))
Now, notice that (y - 4) appears as both a numerator and a denominator. Thus, we can cancel it out:
(y - 4)(y + 4) * (y + 1) / (y * (y - 4)) = (y + 4) * (y + 1) / y
Therefore, the simplified expression is:
(y + 4)(y + 1) / y