Answer:
f(x) = [tex]\frac{1}{4}[/tex] x³ - 3x - 4
Step-by-step explanation:
Given that x = - 2 has multiplicity 2 and x = 4 are zeros then
(x + 2)² and (x - 4) are the factors of the polynomial and the polynomial is the product of the factors, thus
f(x) = a(x + 2)²(x - 4) ← a is a multiplier
To find a substitute (0, 16) into the equation
16 = a(4)(16), thus
64a = 16
a = [tex]\frac{16}{64}[/tex] = [tex]\frac{1}{4}[/tex]
f(x) = [tex]\frac{1}{4}[/tex](x + 2)²(x - 4) ← expand factors
= [tex]\frac{1}{4}[/tex](x² + 4x + 4)(x - 4)
= [tex]\frac{1}{4}[/tex](x³ + 4x² + 4x - 4x² - 16x - 16)
= [tex]\frac{1}{4}[/tex](x³- 12x - 16)
= [tex]\frac{1}{4}[/tex] x³ - 3x - 4
