Question

How do you write a polynomials of least degree with integer coefficients that has the given zeros in expanded form?

Answer 1

`x^3-2x^2-8x+16`

Explanation:

For polynomials, a radical zero cannot occur in isolation; if one of the zeros is `2sqrt(2)`, then the conjugate zero `-2sqrt(2)` must also be present. Thus, we know there are 3 factors:

`(x-2)(x-2sqrt(2))(x+2sqrt(2))`

By multiplying this out we can derive the polynomial. It can be helpful to multiply out the final 2 terms so that the radical is removed:

`(x-2sqrt(2))(x+2sqrt(2)) = x^2+2xsqrt(2)-2xsqrt(2)-4*sqrt(4)`

`= x^2-8`

Finally:

`(x-2)(x^2-8) = x^3 -8x - 2x^2 + 16`

`= x^3-2x^2-8x+16`