find a polynomial

Question

find a polynomial with the zeros -2(multiplicity 2) and 2( multiplicity 2) that passes through the graph (-3,75)

Answer 1

through (-3, 75)

a(x+2)^2(x-2)^2

75 = a(-5)^2)(-1)^2) = 25a

a = 75/25 = 3

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

=3(x^4 -4x^3 +4x^2 +4x^3 -16x^2 +16x +4x^2 -16x +16)

= 3x^4 -24x^2 +48

Answer 2

Let's temporarily forget about the (-3, 75) and begin by satisfying the rest of the problem.

The polynomial

f(x) = (x+2)(x+2)(x-2)(x-2) = x⁴ - 8x² + 16

has the required zeroes and multiplicities.

Its value at x=-3 is

f(-3) = 25

But we want the polynomial's value at x=-3 to be three times as great (75, not 25).

So multiply the polynomial by 3:

3x⁴ - 24x² + 48

All done. That's the required polynomial. Thanks.

Answer 3

To have such zeros and multiplicities, the polynomial in factored form will be:

f(x) = a(x+2)²(x-2)²

Determine a, by plugging in the given point (x,y) = (-3,75):

75 = a(-3+2)²(-3-2)²

75 = a(-1)²(-5)²

75 = 25a

a = 3

The polynomial is: f(x) = 3(x+2)²(x-2)²

Multiply it out if necessary.