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 <math><mstyle displaystyle="true"><mn><mn>0.</mn><mover accent="true"><mn>3</mn><mo>‾</mo></mover></mn></mstyle></math> and simplify the expression. In this case, the expression is equal to <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> so <math><mstyle displaystyle="true"><mn><mn>0.</mn><mover accent="true"><mn>3</mn><mo>‾</mo></mover></mn></mstyle></math> is a root of the polynomial.

Substitute <math><mstyle displaystyle="true"><mn><mn>0.</mn><mover accent="true"><mn>3</mn><mo>‾</mo></mover></mn></mstyle></math> into the polynomial.

Raise <math><mstyle displaystyle="true"><mn><mn>0.</mn><mover accent="true"><mn>3</mn><mo>‾</mo></mover></mn></mstyle></math> to the power of <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn><mn>0.</mn><mover accent="true"><mn>037</mn><mo>‾</mo></mover></mn></mstyle></math> .

Raise <math><mstyle displaystyle="true"><mn><mn>0.</mn><mover accent="true"><mn>3</mn><mo>‾</mo></mover></mn></mstyle></math> to the power of <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mo>-</mo><mn>16</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn><mn>0.</mn><mover accent="true"><mn>1</mn><mo>‾</mo></mover></mn></mstyle></math> .

Subtract <math><mstyle displaystyle="true"><mn><mn>1.</mn><mover accent="true"><mn>7</mn><mo>‾</mo></mover></mn></mstyle></math> from <math><mstyle displaystyle="true"><mn><mn>0.</mn><mover accent="true"><mn>1</mn><mo>‾</mo></mover></mn></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mo>-</mo><mn>103</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn><mn>0.</mn><mover accent="true"><mn>3</mn><mo>‾</mo></mover></mn></mstyle></math> .

Subtract <math><mstyle displaystyle="true"><mn><mn>34.</mn><mover accent="true"><mn>3</mn><mo>‾</mo></mover></mn></mstyle></math> from <math><mstyle displaystyle="true"><mo>-</mo><mn><mn>1.</mn><mover accent="true"><mn>6</mn><mo>‾</mo></mover></mn></mstyle></math> .

Add <math><mstyle displaystyle="true"><mo>-</mo><mn>36</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>36</mn></mstyle></math> .

Since <math><mstyle displaystyle="true"><mn><mn>0.</mn><mover accent="true"><mn>3</mn><mo>‾</mo></mover></mn></mstyle></math> is a known root, divide the polynomial by <math><mstyle displaystyle="true"><mn>3</mn><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.

Divide <math><mstyle displaystyle="true"><mn>3</mn><msup><mrow><mi>x</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>-</mo><mn>16</mn><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>-</mo><mn>103</mn><mi>x</mi><mo>+</mo><mn>36</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>3</mn><mi>x</mi><mo>-</mo><mn>1</mn></mstyle></math> .

Write <math><mstyle displaystyle="true"><mn>3</mn><msup><mrow><mi>x</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>-</mo><mn>16</mn><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>-</mo><mn>103</mn><mi>x</mi><mo>+</mo><mn>36</mn></mstyle></math> as a set of factors.

Factor <math><mstyle displaystyle="true"><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>-</mo><mn>5</mn><mi>x</mi><mo>-</mo><mn>36</mn></mstyle></math> using the AC method.

Consider the form <math><mstyle displaystyle="true"><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mi>b</mi><mi>x</mi><mo>+</mo><mi>c</mi></mstyle></math> . Find a pair of integers whose product is <math><mstyle displaystyle="true"><mi>c</mi></mstyle></math> and whose sum is <math><mstyle displaystyle="true"><mi>b</mi></mstyle></math> . In this case, whose product is <math><mstyle displaystyle="true"><mo>-</mo><mn>36</mn></mstyle></math> and whose sum is <math><mstyle displaystyle="true"><mo>-</mo><mn>5</mn></mstyle></math> .

Write the factored form using these integers.

Remove unnecessary parentheses.

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