Question

Work with your classmates to find some integer values that make the equation below true.In your first post, explain in detail the steps you would take to find these values. If you can find any values, include them in your post.

Answer 1

Given:

`(x^2+3x-4)/(x^2+7x+12)`
`\begin{gathered} (x^2+3x-4)/(x^2+7x+12)=(x^2-x+4x-4)/(x^2+3x+4x+12) \\ Factorizing \\ (x(x-1)+4(x-1))/(x(x+3)+4(x+3))=((x+4)(x-1))/((x+4)(x+3)) \\ =(x-1)/(x+3) \end{gathered}`

To make the condition equal,

`\begin{gathered} (x-1)/(x+3)/(x-1)/(x+1)=(x+1)/(x+3) \\ \\ Hence,\text{ the middle term for the expression to be true is:} \\ (x-1)/(x+1) \end{gathered}`

Hence, the polynomial that can be made for the middle expression is any value multiplied to the numerator and denominator that makes it always true.

Thus,

`\begin{gathered} ((x-1))/((x+1))*((x+4))/((x+4)) \\ =((x-1)(x+4))/((x+1)(x+4)) \\ =(x^2+4x-x-4)/(x^2+4x+x+4) \\ =(x^2+3x-4)/(x^2+5x+4) \end{gathered}`

Therefore,