Question

If p(x) is a cubic polynomial with zeros at -3,-1, and 2, and p(0)=6, find p(x).

Answer 1

`p(x)=-x^3-2x^2+5x+6.`

Explanation:

If a cubic polynomial has zeros
`x_1,\ x_2,\ x_3,` then it has a form

`p(x)=a(x-x_1)(x-x_2)(x-x_3).`

In your case, zeros are -3, -1 and 2, then the polynomial is

`p(x)=a(x-(-3))(x-(-1))(x-2),\\ \\p(x)=a(x+3)(x+1)(x-2).`

If
`p(0)=6,` then

`p(0)=a(0+3)(0+1)(0-2)=6,\\ \\-6a=6,\\ \\a=-1`

and

`p(x)=-(x+3)(x+1)(x-2)=-x^3-2x^2+5x+6.`