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.

Do you know how to Factor f(x)=3x^3-16x^2-103x+36? If not, you can write to our math experts in our application. The best solution for your task you can find above on this page.

Name | one billion one hundred four million two hundred thirteen thousand two hundred twenty-two |
---|

- 1104213222 has 32 divisors, whose sum is
**2296735200** - The reverse of 1104213222 is
**2223124011** - Previous prime number is
**29**

- Is Prime? no
- Number parity even
- Number length 10
- Sum of Digits 18
- Digital Root 9

Name | five hundred fifty-nine million ninety-one thousand six hundred four |
---|

- 559091604 has 32 divisors, whose sum is
**1678152960** - The reverse of 559091604 is
**406190955** - Previous prime number is
**2089**

- Is Prime? no
- Number parity even
- Number length 9
- Sum of Digits 39
- Digital Root 3

Name | four hundred forty-six million six hundred forty-four thousand one hundred fifty-four |
---|

- 446644154 has 16 divisors, whose sum is
**797315616** - The reverse of 446644154 is
**451446644** - Previous prime number is
**11**

- Is Prime? no
- Number parity even
- Number length 9
- Sum of Digits 38
- Digital Root 2