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 .

Multiply by .

One to any power is one.

Raise to the power of .

Cancel the common factor of .

Factor out of .

Factor out of .

Cancel the common factor.

Rewrite the expression.

Combine and .

Multiply .

Multiply by .

Combine and .

Combine fractions.

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.

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 .

Factor out of .

Factor out of .

Factor out of .

Factor out of .

Factor.

Factor by grouping.

For a polynomial of the form , rewrite the middle term as a sum of two terms whose product is and whose sum is .

Factor out of .

Rewrite as plus

Apply the distributive property.

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

Factor out the greatest common factor (GCF) from each group.

Factor the polynomial by factoring out the greatest common factor, .

Remove unnecessary parentheses.

Divide each term in by .

Simplify .

Cancel the common factor of .

Cancel the common factor.

Divide by .

Expand using the FOIL Method.

Apply the distributive property.

Apply the distributive property.

Apply the distributive property.

Simplify and combine like terms.

Simplify each term.

Multiply by .

Multiply by .

Multiply by .

Multiply by .

Multiply by .

Add and .

Divide by .

For a polynomial of the form , rewrite the middle term as a sum of two terms whose product is and whose sum is .

Factor out of .

Rewrite as plus

Apply the distributive property.

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

Factor out the greatest common factor (GCF) from each group.

Factor the polynomial by factoring out the greatest common factor, .

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 .

Add to 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 .

The final solution is all the values that make true.

Find the Roots/Zeros Using the Rational Roots Test 60x^2-57x-18