Find an equation for fx the polynomail of smallest degree wi

Find an equation for f(x), the polynomail of smallest degree with real coefficients such that f(x) breaks through the x-axis at -5, bounces off of the x-axis at -1, has comples roots of 4+2i and -4+4i and passes though the point (0,4).

Solution

f(x) breaks through the x-axis at -5 ----> zero at x=-5 ---> (x+5)

bounces off of the x-axis at -1 -----> zero of multiplicty 2 at x=-1 ----> (x+1)

has comples roots of 4+2i ------> there would be another complex conjugate root x= 4 -2i

and -4+4i ------> there would be another complex conjugate root x= -4 -4i

So, f(x) = a(x+5)(x+1)^2(x -4 -2i)(x- 4 +2i)(x -4-4i)(x-4 +4i)

= a(x+5)(x+1)^2( x^2 -4x +2ix -4x +16 -8i -2ix +8i+ 4)(x^2-4x + 4ix -4x +16 -16i -4ix+16i+16)

= a(x+5)(x+1)^2( x^2 -8x +20)(x^2 -8x +32)

and passes though the point (0,4). find, a , :

4 = a( 5*1*20*32)

1 = a*25*32

a = 1/800

So, f(x) = (1/800)(x+5)(x+1)^2( x^2 -8x +20)(x^2 -8x +32)