What are the rational roots of the polynomial f (x) = 20x^4+x^3+8x^2+x-12

20x^4 + x^3 + 8x^2 + x - 12

1. Arrange the polynomial in descending order (already done)

2. Write down the factors of the constant term (which is 12 here) : 1, 2, 3, 4, 6, 12, - 1, - 2, - 3, - 4, - 6, - 12. These will be known as the P values.

3. Write down the factors of the leading coefficient (which is 20 here) : 1, 2, 4, 5, 10, 20, - 1, - 2, - 4, - 5, - 20. These will be known as the Q values.

4. Write every possible P/Q value. Reduce the fractions, then cross out the duplicates.

*Note: Each of these fractions should be preceeded by a + - sign indicating positive and negative. I am leaving them off for brevity.

-write all values with P=1 on top

1/1, 1/2, 1/3, 1/4, 1/5, 1/10, 1/20

reduce

1, 1/2, 1/3, 1/4, 1/5, 1/10, 1/20

-write all values with P=2 on top

2/1, 2/2, 2/3, 2/4, 2/5, 2/10, 2/20

reduce

2, 1, 2/3, 1/2, 2/5, 1/5, 1/10

-write all values with P=3 on top

3/1, 3/2, 3/3, 3/4, 3/5, 3/10, 3/20

reduce

3, 3/2, 1, 3/4, 3/5, 3/10, 3/20

-write all values with P=4 on top

4/1, 4/2, 4/3, 4/4, 4/5, 4/10, 4/20

reduce

4, 2, 4/3, 1, 4/5, 2/5, 1/5

-write all values with P=6 on top

6/1, 6/2, 6/3, 6/4, 6/5, 6/10, 6/20

reduce

6, 3, 2, 3/2, 6/5, 3/5, 3/10

-write all values with P=12 on top

12/1, 12/2, 12/3, 12/4, 12/5, 12/10, 12/20

reduce

12, 6, 4, 3, 12/5, 6/5, 3/5

These were all the possible solutions (don't forget YOU need to add the + - in front of all the fractions). We now need to remove all of the duplicates

1, 1/2, 1/3, 1/4, 1/5, 1/10, 1/20

2, 1, 2/3, 1/2, 2/5, 1/5, 1/10

3, 3/2, 1, 3/4, 3/5, 3/10, 3/20

4, 2, 4/3, 1, 4/5, 2/5, 1/5

6, 3, 2, 3/2, 6/5, 3/5, 3/10

12, 6, 4, 3, 12/5, 6/5, 3/5

The final set of solutions:

1, 1/2, 1/3, 1/4, 1/5, 1/10, 1/20

2, 2/3, 2/5,

3, 3/2, 3/4, 3/5, 3/10, 3/20

4, 4/3, 4/5,

6, 6/5,

12, 12/5

when you write your solution set write + - 1, + - 1/2 + - 1/3 ... + - 12/5 as all solutions are positive and negative