find a polynomial function with zeros -2 (multiplicity 2) and 2 (multiplicity 2), whose graph passes through the point (-4,432)

Question

polynomial function answer simplified

Answer 1

P(x) = a(x + 2)²(x - 2)²

432 = a(-4 + 2)²(-4 - 2)²

Can you solve for a and answer?

Answer 2

If the roots of a polynomial are -2, -2, 2, and 2 then the factors of that polynomial are (x + 2), (x + 2), (x - 2), and (x - 2) however the polynomial can have a multiplier out front and still have the same roots:

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

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

P(x) = 14(x + 2)(x + 2)(x - 2)(x - 2)

P(x) = -100(x + 2)(x + 2)(x - 2)(x - 2)

These all have the same roots (-2, -2, 2, and 2).

To find out what the CORRECT multiplier is, we use the letter "a" and plug in the given point (-4, 432) and solve for "a":

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

432 = a(-4 + 2)(-4 + 2)(-4 - 2)(-4 - 2)

432 = a(-2)(-2)(-6)(-6)

432 = 144a

a = 432/144 = 3 therefore the polynomial is:

P(x) = 3(x + 2)(x + 2)(x - 2)(x - 2) or

P(x) = 3(x + 2)²(x - 2)²