Find a formula for P(x).

Question

The polynomial P(x) of degree 4 has

1. a root of multiplicity 2 at x = 4
2. a root of multiplicity 1 at x = 0 and at x = -3
3. It goes through the point (5,8)

Find a formula for P(x).

Answer 1

ax(x-4)^2(x+3) through (5,8)

= ax(x^2-4x+4)(x+3)

=ax(x^3 -x^2-8x +12)

=a(x^4 -x^3 -8x^2 +12x), plug in the point to find a value for "a" the leading term's coefficient

8=a(625 -125 -200+60) = 360a

a = 360/8 = 45

P(x) = 45x^4 -45x^3 -360x +545