Question

The polynomial of degree 3, P(x), has a root of. multiplicity 2 at x=1 and a roof of multiplicity 1 at x= -3. the y- intercept is y= 2.1. Find a formula for P(x)​

Answer 1

`P(x)` has degree 3, so it takes the general form

`P(x)=ax^3+bx^2+cx+d`

It has a root
`x=1` of multiplicity 2, which means
`(x-1)^2` divides
`P(x)` exactly, and it has a root of
`x=-3` of multiplicity 1 so that
`x+3` also is a factor. So

`P(x)=a(x-1)^2(x+3)`

Expanding this gives

`P(x)=a(x^3+x^2-5x+3)=ax^3+ax^2-5ax+3a`

`\implies\begin{cases}a=b\\-5a=c\\3a=d\end{cases}`

The
`y`-intercept occurs for
`x=0`, for which we have

`P(0)=d=2.1`

Then

`3a=d\implies a=0.7`

`a=b\implies b=0.7`

`-5a=c\implies c=-0.14`

So we have

`\boxed{P(x)=0.7x^3+0.7x^2-0.14x+2.1}`

Answer 2

4

Explanation:

on edge.