How to Write Polynomial Functions When Given Zeros

By Sreela Datta
Writing the mathematical process makes the task easier.
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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.

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.

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.

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).

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.

About the Author

Sreela Datta has been writing on technical and scientific topics since 1995. Her articles have been published in the "Gale Encyclopedia of Science" and she has developed teacher training materials for XAMOnline. Datta earned a Ph.D. in physics from Iowa State University.