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.

Use the power rule to distribute the exponent.

Apply the product rule to .

Apply the product rule to .

Raise to the power of .

One to any power is one.

Raise to the power of .

Cancel the common factor of .

Move the leading negative in into the numerator.

Factor out of .

Cancel the common factor.

Rewrite the expression.

Move the negative in front of the fraction.

Use the power rule to distribute the exponent.

Apply the product rule to .

Apply the product rule to .

Raise to the power of .

Multiply by .

One to any power is one.

Raise to the power of .

Combine and .

Move the negative in front of the fraction.

Multiply .

Multiply by .

Combine and .

Simplify terms.

Combine fractions with similar denominators.

Subtract from .

Cancel the common factor of and .

Factor out of .

Cancel the common factors.

Factor out of .

Cancel the common factor.

Rewrite the expression.

Move the negative in front of the fraction.

Combine fractions with similar denominators.

Simplify the expression.

Add and .

Divide by .

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.

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.

Factor out of .

Factor out of .

Factor out of .

Factor out of .

Factor out of .

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 .

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.

If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .

Set the first factor equal to .

Subtract from both sides of the equation.

Divide each term by and simplify.

Divide each term in by .

Cancel the common factor of .

Cancel the common factor.

Divide by .

Move the negative in front of the fraction.

Set the next factor equal to .

Use the quadratic formula to find the solutions.

Substitute the values , , and into the quadratic formula and solve for .

Simplify.

Simplify the numerator.

Raise to the power of .

Multiply by .

Multiply by .

Add and .

Multiply by .

Simplify the expression to solve for the portion of the .

Simplify the numerator.

Raise to the power of .

Multiply by .

Multiply by .

Add and .

Multiply by .

Change the to .

Simplify the expression to solve for the portion of the .

Simplify the numerator.

Raise to the power of .

Multiply by .

Multiply by .

Add and .

Multiply by .

Change the to .

The final answer is the combination of both solutions.

The final solution is all the values that make true.

The result can be shown in multiple forms.

Exact Form:

Decimal Form:

Find the Roots/Zeros Using the Rational Roots Test 2x^3-5x^2-5x-1=0