Question

What is the polynomial function `f` of least degree that has rational coefficients, a leading coefficient of 2, and the zeros -4, 0, 2, 4?

Answer 1

`f(x)=2x^4-4x^3-32x^2+64x`

Explanation:

Given a function `p` and zeroes `x_1,x_2,...,x_n`, we can write `p(x)=a(x-x_1)(x-x_2)...(x-x_n)` where `a` is the leading coefficient or dilation factor. Knowing this, we can write the equation of `f` as `f(x)=2(x-(-4))(x-0)(x-2)(x-4)`.
`f(x)=2(x+4)(x)(x-2)(x-4)`
`f(x)=2x(x^2-16)(x-2)`
`f(x)=2x(x^3-2x^2-16x+32)`
`:.f(x)=2x^4-4x^3-32x^2+64x`