How To Write Polynomial Functions When Given Zeros

The zeros of a polynomial function of x are the values of x that make the function zero. For example, the polynomial x^3 – 4x^2 + 5x – 2 has zeros x = 1 and x = 2. When x = 1 or 2, the polynomial equals zero. One way to find the zeros of a polynomial is to write in its factored form. The polynomial x^3 – 4x^2 + 5x – 2 can be written as (x – 1)(x – 1)(x – 2) or ((x – 1)^2)(x – 2). Just by looking at the factors, you can tell that setting x = 1 or x = 2 will make the polynomial zero. Notice that the factor x – 1 occurs twice. Another way to say this is that the multiplicity of the factor is 2. Given the zeros of a polynomial, you can very easily write it — first in its factored form and then in the standard form.

Step 1

Subtract the first zero from x and enclose it in parentheses. This is the first factor. For example if a polynomial has a zero that is -1, the corresponding factor is x – (-1) = x + 1.

Step 2

Raise the factor to the power of the multiplicity. For instance, if the zero -1 in the example has a multiplicity of two, write the factor as (x + 1)^2.

Step 3

Repeat Steps 1 and 2 with the other zeros and add them as further factors. For instance, if the example polynomial has two more zeros, -2 and 3, both with multiplicity 1, two more factors — (x + 2) and (x – 3) — must be added to the polynomial. The final form of the polynomial is then ((x + 1)^2)(x + 2)(x – 3).

Step 4

Multiply out all the factors using the FOIL (First Outer Inner Last) method to get the polynomial in the standard form. In the example, first multiply (x + 2)(x – 3) to get x^2 + 2x – 3x – 6 = x^2 – x – 6. Then multiply this with another factor (x + 1) to get (x^2 – x – 6)(x + 1) = x^3 +x^2 – x^2 – x – 6x – 6 = x^3 – 7x – 6. Finally, multiply this with the last factor (x + 1) to get (x^3 – 7x – 6)(x + 1) = x^4 + x^3 -7x^2 – 7x – 6x – 6 = x^4 + x^3 – 7x^2 – 13x – 6. This is the standard form of the polynomial.