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

Answer:

4

Step-by-step explanation:

on edge.