Graphs of Polynomial Functions

Question

9

The polynomial of degree 4, , has a root of multiplicity 2 at  and roots of multiplicity 1 at  and . It goes through the point .

Find a formula for .

Answer 1

The degree of your polynomial tells you exactly how many complex roots it has, and thus the maximum number of real roots. In your case, P(x) is degree 4 so it could have 4 real roots. In fact, the problem tells us that it has 4 real roots, one of which has multiplicity 2. Any polynomial equation can be found by multiplying the factors that can be obtained from the roots. For instance since x=-2 is a root, then (x+2) is a factor of P(x). Since x=1 is a root of multiplicity 2, then (x-1)² is two factors.

Thus P(x) = c*x(x+2)(x-1)²

c is some real number that you can find by plugging in the point (5,280) and solving for c.