Question

Reduce the rational expression to lowest terms. If it is already in lowest terms, enter the expression in the answer box. Also, specify any restrictions on the variable.y³ - 2y² - 9y + 18/y² + y - 6Rational expression in lowest terms:Variable restrictions for the original expression: y

Answer 1

`\begin{gathered} \text{ Rational expression in lowest terms: }y-3 \\ \\ \text{ Variable restrictions for the original expression: }y\\e2,-3 \end{gathered}`

Step-by-step explanation

We want to reduce the rational expression to the lowest terms:

`(y^3-2y^2-9y+18)/(y^2+y-6)`

First, let us factor the denominator of the expression:

`\begin{gathered} y^2+y-6 \\ \\ y^2+3y-2y-6 \\ \\ y(y+3)-2(y+3) \\ \\ (y-2)(y+3) \end{gathered}`

Now, we can test if the factors in the denominator are also the factors in the numerator.

To do this for (y - 2), substitute y = 2 in the numerator. If it is equal to 0, then, it is a factor:

`\begin{gathered} (2)^3-2(2)^2-9(2)+18 \\ \\ 8-8-18+18 \\ \\ 0 \end{gathered}`

Since it is equal to 0, (y - 2) is a factor. Now, let us divide the numerator by (y -2):

We have simplified the numerator and now, we can factorize by the difference of two squares:

`\begin{gathered} y^2-9 \\ \\ y^2-3^2 \\ \\ (y-3)(y+3) \end{gathered}`

Therefore, the simplified expression is:

`((y-2)(y-3)(y+3))/((y-2)(y+3))`

Simplify further by dividing common terms. The expression becomes:

`y-3`

That is the rational expression in the lowest terms.

To find the variable restrictions, set the denominator of the original expression to 0 and solve for y:

`\begin{gathered} y^2+y-6=0 \\ \\ y^2+3y-2y-6=0 \\ \\ y(y+3)-2(y+3)=0 \\ \\ (y-2)(y+3)=0 \\ \\ y=2,\text{ }y=-3 \end{gathered}`

Those are the variable restrictions for the original expression.