Question

Write a polynomial equation of degree 4 that has the following roots: -1 repeated three times and 4.

Answer 1

`\bf \begin{cases} x=-1\to x+1=0\to &(x+1)=0\\ x=-1\to x+1=0\to &(x+1)=0\\ x=-1\to x+1=0\to &(x+1)=0\\ x=4\to x-4=0\to &(x-4)=0 \end{cases} \\\\\\ (x+1)(x+1)(x+1)(x-4)=0\implies (x+1)^3(x-4)=0 \\\\\\ (x+1)^3(x-4)=\textit{original polynomial}`

those are the roots, multiply them, to get the polynomial

Answer 2

`f(x)= (x+1)^3(x-4)`

Explanation:

a polynomial equation of degree 4 that has the following roots: -1 repeated three times and 4.

-1 is repeated three times

so roots are -1,-1, -1,4

We write the roots in factor form

If 'a' is a root then (x-a) is a factor

roots are -1,-1, -1,4

Factors are (x+1)(x+1)(x+1)(x-4)

`f(x)= (x+1)(x+1)(x+1)(x-4)`

`f(x)= (x+1)^3(x-4)`