If a polynomial function has integer coefficients, then every rational zero will have the form <math><mstyle displaystyle="true"><mfrac><mrow><mi>p</mi></mrow><mrow><mi>q</mi></mrow></mfrac></mstyle></math> where <math><mstyle displaystyle="true"><mi>p</mi></mstyle></math> is a factor of the constant and <math><mstyle displaystyle="true"><mi>q</mi></mstyle></math> is a factor of the leading coefficient.

Find every combination of <math><mstyle displaystyle="true"><mo>±</mo><mfrac><mrow><mi>p</mi></mrow><mrow><mi>q</mi></mrow></mfrac></mstyle></math> . 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 <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> , which means it is a root.

Simplify each term.

Raise <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> to the power of <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> .

Raise <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> to the power of <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mn>5</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mo>-</mo><mn>139</mn></mstyle></math> by <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> .

Simplify by adding and subtracting.

Add <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>5</mn></mstyle></math> .

Add <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>139</mn></mstyle></math> .

Subtract <math><mstyle displaystyle="true"><mn>143</mn></mstyle></math> from <math><mstyle displaystyle="true"><mn>143</mn></mstyle></math> .

Since <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> is a known root, divide the polynomial by <math><mstyle displaystyle="true"><mi>x</mi><mo>+</mo><mn>1</mn></mstyle></math> 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 <math><mstyle displaystyle="true"><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mstyle></math> is put into the first position of the result area (below the horizontal line).

Multiply the newest entry in the result <math><mstyle displaystyle="true"><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mstyle></math> by the divisor <math><mstyle displaystyle="true"><mrow><mo>(</mo><mo>-</mo><mn>1</mn><mo>)</mo></mrow></mstyle></math> and place the result of <math><mstyle displaystyle="true"><mrow><mo>(</mo><mo>-</mo><mn>1</mn><mo>)</mo></mrow></mstyle></math> under the next term in the dividend <math><mstyle displaystyle="true"><mrow><mo>(</mo><mn>5</mn><mo>)</mo></mrow></mstyle></math> .

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 <math><mstyle displaystyle="true"><mrow><mo>(</mo><mn>4</mn><mo>)</mo></mrow></mstyle></math> by the divisor <math><mstyle displaystyle="true"><mrow><mo>(</mo><mo>-</mo><mn>1</mn><mo>)</mo></mrow></mstyle></math> and place the result of <math><mstyle displaystyle="true"><mrow><mo>(</mo><mo>-</mo><mn>4</mn><mo>)</mo></mrow></mstyle></math> under the next term in the dividend <math><mstyle displaystyle="true"><mrow><mo>(</mo><mo>-</mo><mn>139</mn><mo>)</mo></mrow></mstyle></math> .

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 <math><mstyle displaystyle="true"><mrow><mo>(</mo><mo>-</mo><mn>143</mn><mo>)</mo></mrow></mstyle></math> by the divisor <math><mstyle displaystyle="true"><mrow><mo>(</mo><mo>-</mo><mn>1</mn><mo>)</mo></mrow></mstyle></math> and place the result of <math><mstyle displaystyle="true"><mrow><mo>(</mo><mn>143</mn><mo>)</mo></mrow></mstyle></math> under the next term in the dividend <math><mstyle displaystyle="true"><mrow><mo>(</mo><mo>-</mo><mn>143</mn><mo>)</mo></mrow></mstyle></math> .

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.

Use the quadratic formula to find the solutions.

Substitute the values <math><mstyle displaystyle="true"><mi>a</mi><mo>=</mo><mn>1</mn></mstyle></math> , <math><mstyle displaystyle="true"><mi>b</mi><mo>=</mo><mn>4</mn></mstyle></math> , and <math><mstyle displaystyle="true"><mi>c</mi><mo>=</mo><mo>-</mo><mn>143</mn></mstyle></math> into the quadratic formula and solve for <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> .

Simplify.

Simplify the numerator.

Raise <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> to the power of <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mo>-</mo><mn>143</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mo>-</mo><mn>4</mn></mstyle></math> by <math><mstyle displaystyle="true"><mo>-</mo><mn>143</mn></mstyle></math> .

Add <math><mstyle displaystyle="true"><mn>16</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>572</mn></mstyle></math> .

Rewrite <math><mstyle displaystyle="true"><mn>588</mn></mstyle></math> as <math><mstyle displaystyle="true"><msup><mrow><mn>14</mn></mrow><mrow><mn>2</mn></mrow></msup><mo>⋅</mo><mn>3</mn></mstyle></math> .

Factor <math><mstyle displaystyle="true"><mn>196</mn></mstyle></math> out of <math><mstyle displaystyle="true"><mn>588</mn></mstyle></math> .

Rewrite <math><mstyle displaystyle="true"><mn>196</mn></mstyle></math> as <math><mstyle displaystyle="true"><msup><mrow><mn>14</mn></mrow><mrow><mn>2</mn></mrow></msup></mstyle></math> .

Pull terms out from under the radical.

Multiply <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Simplify <math><mstyle displaystyle="true"><mfrac><mrow><mo>-</mo><mn>4</mn><mo>±</mo><mn>14</mn><msqrt><mn>3</mn></msqrt></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> .

Simplify the expression to solve for the <math><mstyle displaystyle="true"><mo>+</mo></mstyle></math> portion of the <math><mstyle displaystyle="true"><mo>±</mo></mstyle></math> .

Simplify the numerator.

Raise <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> to the power of <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mo>-</mo><mn>143</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mo>-</mo><mn>4</mn></mstyle></math> by <math><mstyle displaystyle="true"><mo>-</mo><mn>143</mn></mstyle></math> .

Add <math><mstyle displaystyle="true"><mn>16</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>572</mn></mstyle></math> .

Rewrite <math><mstyle displaystyle="true"><mn>588</mn></mstyle></math> as <math><mstyle displaystyle="true"><msup><mrow><mn>14</mn></mrow><mrow><mn>2</mn></mrow></msup><mo>⋅</mo><mn>3</mn></mstyle></math> .

Factor <math><mstyle displaystyle="true"><mn>196</mn></mstyle></math> out of <math><mstyle displaystyle="true"><mn>588</mn></mstyle></math> .

Rewrite <math><mstyle displaystyle="true"><mn>196</mn></mstyle></math> as <math><mstyle displaystyle="true"><msup><mrow><mn>14</mn></mrow><mrow><mn>2</mn></mrow></msup></mstyle></math> .

Pull terms out from under the radical.

Multiply <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Simplify <math><mstyle displaystyle="true"><mfrac><mrow><mo>-</mo><mn>4</mn><mo>±</mo><mn>14</mn><msqrt><mn>3</mn></msqrt></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> .

Change the <math><mstyle displaystyle="true"><mo>±</mo></mstyle></math> to <math><mstyle displaystyle="true"><mo>+</mo></mstyle></math> .

Simplify the expression to solve for the <math><mstyle displaystyle="true"><mo>-</mo></mstyle></math> portion of the <math><mstyle displaystyle="true"><mo>±</mo></mstyle></math> .

Simplify the numerator.

Raise <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> to the power of <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mo>-</mo><mn>143</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mo>-</mo><mn>4</mn></mstyle></math> by <math><mstyle displaystyle="true"><mo>-</mo><mn>143</mn></mstyle></math> .

Add <math><mstyle displaystyle="true"><mn>16</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>572</mn></mstyle></math> .

Rewrite <math><mstyle displaystyle="true"><mn>588</mn></mstyle></math> as <math><mstyle displaystyle="true"><msup><mrow><mn>14</mn></mrow><mrow><mn>2</mn></mrow></msup><mo>⋅</mo><mn>3</mn></mstyle></math> .

Factor <math><mstyle displaystyle="true"><mn>196</mn></mstyle></math> out of <math><mstyle displaystyle="true"><mn>588</mn></mstyle></math> .

Rewrite <math><mstyle displaystyle="true"><mn>196</mn></mstyle></math> as <math><mstyle displaystyle="true"><msup><mrow><mn>14</mn></mrow><mrow><mn>2</mn></mrow></msup></mstyle></math> .

Pull terms out from under the radical.

Multiply <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Simplify <math><mstyle displaystyle="true"><mfrac><mrow><mo>-</mo><mn>4</mn><mo>±</mo><mn>14</mn><msqrt><mn>3</mn></msqrt></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> .

Change the <math><mstyle displaystyle="true"><mo>±</mo></mstyle></math> to <math><mstyle displaystyle="true"><mo>-</mo></mstyle></math> .

The final answer is the combination of both solutions.

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

These are the roots (zeros) of the polynomial <math><mstyle displaystyle="true"><msup><mrow><mi>x</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>+</mo><mn>5</mn><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>-</mo><mn>139</mn><mi>x</mi><mo>-</mo><mn>143</mn></mstyle></math> .

The result can be shown in multiple forms.

Exact Form:

Decimal Form:

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