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 |
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 | 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)
