# Finite Math Examples

If a polynomial function has integer coefficients, then every rational zero will have the form where is a factor of the constant and is a factor of the leading coefficient.

Find every combination of . These are the possible roots of the polynomial function.

Substitute the possible roots one by one into the polynomial to find the actual roots. Simplify to check if the value is , which means it is a root.

Simplify each term.

Remove parentheses.

Raise to the power of .

Multiply by .

Simplify by adding and subtracting.

Subtract from .

Add and .

Since is a known root, divide the polynomial by to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

Place the numbers representing the divisor and the dividend into a division-like configuration.

The first number in the dividend is put into the first position of the result area (below the horizontal line).

Multiply the newest entry in the result by the divisor and place the result of under the next term in the dividend .

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

Multiply the newest entry in the result by the divisor and place the result of under the next term in the dividend .

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

All numbers except the last become the coefficients of the quotient polynomial. The last value in the result line is the remainder.

Simplify the quotient polynomial.

Add to both sides of the equation.

The polynomial can be written as a set of linear factors.

These are the roots (zeros) of the polynomial .