Question

Suppose f(x) is a degree three polynomial with ⋅ a root of multiplicity 2 at x=1⋅ a root of multiplicity 1 at x=−2⋅ a y-intercept at (0,4).Find a formula for f(x).

Answer 1

Since x=1 is a root of multiplicity 2 and x= -2 is a root with multiplicity 1, our polynomial has the form

`f(x)=A\cdot(x-1)^2\cdot(x+2)`

where A is an unknown constant.

We can find A by substituting into our last result the given point (0,4), that is,

`4=A\cdot(0-1)^2\cdot(0+2)`

which gives

`\begin{gathered} 4=A\cdot(-1)^2\cdot2 \\ \text{then} \\ 4=A\cdot1\cdot2 \\ so \\ 4=2A \end{gathered}`

then, by dividing both sides by 2, we get

`A=2`

Therefore, the answer is:

`f(x)=2(x-1)^2(x+2)`