The polynomial of degree 4, P ( x ) has a root of multiplicity 2 at x = 2 and roots of multiplicity 1 at x = 0 and x = − 2 . It goes through the point ( 5 , 220.5 ) .

Question

P(X)=

Answer 1

If the roots of P(x) are 2, 2, 0, -2 then its factors are (x - 2), (x - 2), (x - 0), and (x + 2) which makes the polynomial:

P(x) = a(x - 2)(x - 2)(x - 0)(x + 2)

Notice that there is an "a" in front of the factors because I can multiply by any number and still have the same factors/roots.

To find out what "a" is, use the given point (5, 220.5) which is a value of (x, y) on the graph. Plug in y = 220.5 and x = 5 and solve for "a":

220.5 = = a(5 - 2)(5 - 2)(5 - 0)(5 + 2)

220.5 = = a(3)(3)(5)(7)

220.5 = 315a

a = 0.7

Then plug that into the function we developed:

P(x) = 0.7(x - 2)(x - 2)(x - 0)(x + 2)

P(x) = 0.7x(x - 2)(x - 2)(x + 2) or

P(x) = 0.7x(x - 2)²(x + 2)

You can multiply it out if desired but I would use this answer.