Question

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

Answer 1

Answer:

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

Step-by-step 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

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