Question

What is the product in simplest form? State any restrictions on the variable. y^2/y-3 * y^2-y-6/y^2+1y

Answer 1

the product of given expression
`(y^2)/(y-3)` and
`(y^2-y-6)/(y^2+y)` is
`(y(y+2))/((y+1)),y\\eq -1`

Step-by-step explanation:

Consider the given expression
`(y^2)/(y-3)` and
`(y^2-y-6)/(y^2+y)`

We have to write the product of given expressions in simplest form,

We solve the two terms separately,

Consider the second expression

`(y^2-y-6)/(y^2+y)`

Taking y common from denominator we have
`(y^2-y-6)/(y(y+1))`

Numerator consist of a quadratic equation,
`{y^2-y-6}`

We can solve it using middle term splitting method,

-y can be written as 2y -3y

Thus, it becomes,

`{y^2-y-6}=y^2+2y-3y-6`

Taking y common from first two terms and -3 common from last two terms we have,

`y^2+2y-3y-6=y(y+2)-3(y+2)=(y-3)(y+2)`

thus, the second expression becomes ,

`(y^2-y-6)/(y(y+1))=((y-3)(y+2))/(y(y+1))`

Now , we multiply the two expression ,we get,

`\Rightarrow (y^2)/(y-3)* (y^2-y-6)/(y^2+y)`

`\Rightarrow (y^2)/(y-3)* ((y-3)(y+2))/(y(y+1))`

Simplifying , we get,

`\Rightarrow y * ((y+2))/((y+1))\\\\\\ \Rightarrow(y(y+2))/((y+1)),y\\eq -1`

Thus, the product of given expression
`(y^2)/(y-3)` and
`(y^2-y-6)/(y^2+y)` is
`(y(y+2))/((y+1)),y\\eq -1`

Answer 2

Answer:

`(y(y+2))/((y+1))`

Step-by-step explanation:
When multiplying fractions, we multiply the numerators together and the denominator together and then we simplify.

The given expression is:

`(y^2)/(y-3) * (y^2-y-6)/(y^2+y)`

Let's consider the second fraction given:

`(y^2-y-6)/(y^2+y)`
We can do the following:
1- factor the numerator
2- take y as a common factor from the denominator
Doing this, we will end up with the following fraction:

`((y-3)(y+2))/(y(y+1))`

The expression now became:

`(y^2)/(y-3) * ((y-3)(y+2))/(y(y+1))`

We can now multiply the numerators together and the denominator together as follows:

`(y^2)/(y-3) * ((y-3)(y+2))/(y(y+1))` =
`(y^2(y-3)(y+2))/(y(y+1)(y-3))`

Finally, we can note that we can perform the following on the resultant expression:
1- divide both the numerator and denominator by (y-3)
2- divide both the numerator and denominator by (y)
Doing this, we will end up with:

`(y(y+2))/((y+1))`

Since no further simplification can be done, therefore, our final expression is:

`(y(y+2))/((y+1))`

Hope this helps :)