Write an exact equation in factored form for the cubic polyn

Write an exact equation, in factored form, for the cubic polynomial whose graph passes through the points (-2.0) and (1, -3) and is tangent to the x-axis at the origin.

Solution

P(x) = ax^3 + bx^2 + cx + d
It has zero at (-2, 0) :

0 = -8a + 4b -2c +d ----(1)

pass through : (1, -3)

-3 = a +b + c + d

Tangent to x axis at origin : (0,0) is a point on P(x)

d =0

P\'(0) =0 ; 0 = 3ax^2 + 2bx +c ; c =0

P(x) = ax^3 + bx^2

So, we have two equation : 0 = -8a + 4b -2c +d -----> 0 = -8a +4b ; b = 2a

and -3 = a +b + c + d ; -3 = a+b ;

So, -3 = a +2a ; a = -1 ; b = -2

So,       P(x) = -x^3 - 2x^2