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 and simplify the expression. In this case, the expression is equal to so is a root of the polynomial.

Substitute into the polynomial.

Raise to the power of .

Multiply by .

Raise to the power of .

Multiply by .

Subtract from .

Raise to the power of .

Multiply by .

Add and .

Multiply by .

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.

Divide by .

Write as a set of factors.

Factor using the rational roots test.

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 and simplify the expression. In this case, the expression is equal to so is a root of the polynomial.

Substitute into the polynomial.

Raise to the power of .

Raise to the power of .

Multiply by .

Subtract from .

Multiply by .

Add and .

Subtract from .

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.

Divide by .

Write as a set of factors.

Remove unnecessary parentheses.

Factor 3x^4-37x^3+111x^2-45x+4