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.
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