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.

Raise to the power of .

Raise to the power of .

Multiply by .

Raise to the power of .

Multiply by .

Multiply by .

Simplify by adding and subtracting.

Subtract from .

Add and .

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.

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.

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, .

Regroup terms.

Factor out of .

Factor out of .

Factor out of .

Factor out of .

Rewrite as .

Let . Substitute for all occurrences of .

Factor using the AC method.

Consider the form . Find a pair of integers whose product is and whose sum is . In this case, whose product is and whose sum is .

Write the factored form using these integers.

Replace all occurrences of with .

Factor out of .

Factor out of .

Factor out of .

Let . Substitute for all occurrences of .

Factor using the perfect square rule.

Rearrange terms.

Rewrite as .

Check the middle term by multiplying and compare this result with the middle term in the original expression.

Simplify.

Factor using the perfect square trinomial rule , where and .

Replace all occurrences of with .

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.

Take the square root of both sides of the equation to eliminate the exponent on the left side.

The complete solution is the result of both the positive and negative portions of the solution.

Rewrite as .

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the to find the first solution.

Next, use the negative value of the to find the second solution.

The complete solution is the result of both the positive and negative portions of the solution.

Set the next factor equal to .

Set the equal to .

Subtract from both sides of the equation.

The final solution is all the values that make true.

Find the Roots/Zeros Using the Rational Roots Test x^4+12x^3+37x^2+12x+36