Find an equation for Px the polynomial of smallest degree wi

Find an equation for P(x), the polynomial of smallest degree with real coefficients in factored form such that P(x) bounces off of the x-axis at -1, breaks through the x-axis at 1. has a complex root 2i and passes through the point (0,8)

Solution

P(x) bounces off of the x-axis at -1, ----- touches at x axis i.e. root of multiplicity 2

= (x+1)^2

breaks through the x-axis at 1 ----- root at x= 1 , (x-1)

has a complex root 2i , there would be another conjugate root = -2i

=(x+2i)(x-2i) =(x^2 +4)

P(x) = k(x+1)^2(x-1)(x^2 +4)

Now find the constant k , using the point (0,8)

8 = k(1)^2(-1)(4) ---->k = -2

So, P(x) = -2(x+1)^2(x-1)(x^2 +4)