X(X-1)=2(X-1)(X+5)(X-4)(X-4)

If a polynomial function has integer coefficients, then every rational zero will have the form pq where p is a factor of the constant and q is a factor of the leading coefficient.

p=±1,±2,±4,±5,±8,±10,±16,±20,±32,±40,±80,±160

q=±1,±2

Find every combination of ±pq. These are the possible roots of the polynomial function.

±1,±12,±2,±4,±5,±52,±8,±10,±16,±20,±32,±40,±80,±160

Substitute the possible roots one by one into the polynomial to find the actual roots. Simplify to check if the value is 0, which means it is a root.

2(1)4-8(1)3-43(1)2+209(1)-160

Simplify each term.

One to any power is one.

2⋅1-8(1)3-43(1)2+209(1)-160

Multiply 2 by 1.

2-8(1)3-43(1)2+209(1)-160

One to any power is one.

2-8⋅1-43(1)2+209(1)-160

Multiply -8 by 1.

2-8-43(1)2+209(1)-160

One to any power is one.

2-8-43⋅1+209(1)-160

Multiply -43 by 1.

2-8-43+209(1)-160

Multiply 209 by 1.

2-8-43+209-160

2-8-43+209-160

Simplify by adding and subtracting.

Subtract 8 from 2.

-6-43+209-160

Subtract 43 from -6.

-49+209-160

Add -49 and 209.

160-160

Subtract 160 from 160.

0

0

0

Since 1 is a known root, divide the polynomial by X-1 to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

2X4-8X3-43X2+209X-160X-1

Place the numbers representing the divisor and the dividend into a division-like configuration.

1 | 2 | -8 | -43 | 209 | -160 |

The first number in the dividend (2) is put into the first position of the result area (below the horizontal line).

1 | 2 | -8 | -43 | 209 | -160 |

2 |

Multiply the newest entry in the result (2) by the divisor (1) and place the result of (2) under the next term in the dividend (-8).

1 | 2 | -8 | -43 | 209 | -160 |

2 | |||||

2 |

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

1 | 2 | -8 | -43 | 209 | -160 |

2 | |||||

2 | -6 |

Multiply the newest entry in the result (-6) by the divisor (1) and place the result of (-6) under the next term in the dividend (-43).

1 | 2 | -8 | -43 | 209 | -160 |

2 | -6 | ||||

2 | -6 |

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

1 | 2 | -8 | -43 | 209 | -160 |

2 | -6 | ||||

2 | -6 | -49 |

Multiply the newest entry in the result (-49) by the divisor (1) and place the result of (-49) under the next term in the dividend (209).

1 | 2 | -8 | -43 | 209 | -160 |

2 | -6 | -49 | |||

2 | -6 | -49 |

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

1 | 2 | -8 | -43 | 209 | -160 |

2 | -6 | -49 | |||

2 | -6 | -49 | 160 |

Multiply the newest entry in the result (160) by the divisor (1) and place the result of (160) under the next term in the dividend (-160).

1 | 2 | -8 | -43 | 209 | -160 |

2 | -6 | -49 | 160 | ||

2 | -6 | -49 | 160 |

1 | 2 | -8 | -43 | 209 | -160 |

2 | -6 | -49 | 160 | ||

2 | -6 | -49 | 160 | 0 |

All numbers except the last become the coefficients of the quotient polynomial. The last value in the result line is the remainder.

2X3+-6X2+(-49)X+160

Simplify the quotient polynomial.

2X3-6X2-49X+160

2X3-6X2-49X+160

Graph each side of the equation. The solution is the x-value of the point of intersection.

X≈-5.03084282,3.54456995,4.48627286

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

(X-1)(X+5.03084282)(X-3.54456995)(X-4.48627286)

These are the roots (zeros) of the polynomial 2X4-8X3-43X2+209X-160.

X=1,-5.03084282,3.54456995,4.48627286

Solve for X X(X-1)=2(X-1)(X+5)(X-4)(X-4)