Question

How to write a polynomial function with rational coefficients in standard form with the given zeros?

Answer 1

If you have as many zeros (z1, z2,... zn) as the degree of the polynomial function (n), then the function equation is given by:

f(x) = k (x - z1) (x - z2) ··· (x - zn)

where k is a constant. To determine this constant you need additional information (e.g. another point of the function).

You might also have less zeros than the degree of the polynomial, because some of them have a multiplicity greater than 1. Let z1,...,zr be the zeros, and m1,...mr be the multiplicity of each zero, where m1+m2+...+mr=n (we assume the sum of all multiplicities equals the degree of the polynomial). In this case, you can use:

$f(x)=k(x-z_1)^{m_1} (x-z_2)^{m_2} \cdots (x-z_r)^{m_r}$

Again, you have an undetermined constant factor.

From any of the two expressions above, you have to apply the distributive property to go on to the standard form.

An example: the zeros of a 3-degree polynomial function are -2, 1 and 3. The function equation is of this form:

$f(x)=k(x+2)(x-1)(x-3) = k(x^3-2x^2-5x+6)$

If you don't have all the zeros, or the sum of multiplicities does not equal the degree of  the polynomial, you have not enough information to write the equation of the function.