Polynomial Formulas

Question

Find a possible formula for a fourth degree polynomial g that has a double zero at 2, g(−6)=0,  g(6)=0, and g(0)=12.

Answer 1

double zero at 2

and points (+/-6,0) are x intercepts

(0,12) is a y intercept

f(x) =a(x-2)^2(x-6)(x+6)

12 = a(4)(-36)

1 = -12a

a = -1/12

f(x) = -(1/12)(x-2)^2(x^2-36)

= -(1/12)(x^2 -4x -4)(x^2-36)

= -(1/12)(x^4 -4x^3 -40x^2 +144x +144)

= -(x^4)/12 + (x^3)/3 +10(x^2)/3 - 12x -12

Answer 2

g(x) = a(x - 2)²(x² - 36); g(0) = a(0 - 2)²(0 - 36) = 12; a = - 1/12.

So, g(x) = -1/12(x - 2)²(x² - 36)