Question

Construct a polynomial function with the given characteristics:

Zeroes : ―5 (multiplicity 2); 2 (multiplicity 1); 4 (multiplicity 1)
Degree : 4
Contains the point : (3, 128)

Answer 1

f(x) = -2(x +5)²(x -2)(x -4)

Explanation:

When a polynomial has a zero of x=p, that means it has a factor of (x-p). The multiplicity of a zero corresponds to the exponent of that factor (the number of times it is a factor). The polynomial can be made to include a specific point by choosing a vertical scale factor. Here, that is represented by 'a'.

Your polynomial will be ...

f(x) = a(x +5)²(x -2)(x -4)

We want f(3) = 128, so ...

128 = a(3 +5)²(3 -2)(3 -4) = -64a

-2 = a . . . . . divide by -64

The function is then ...

f(x) = -2(x +5)²(x -2)(x -4)