William W. answered • 10/17/22
Math and science made easy - learn from a retired engineer
The rational roots theorem states that the POSSIBLE rational roots are the factors of the last term (the constant at the end) divided by the factors of the leading coefficient (and either plus or minus of these).
In this case, the factors of the last term are 34, 17, 2, 1 and the factors of the leading coefficient are 2 and 1. So the possible rational roots are: ± (34/2, 17/2, 2/2, 1/2, 34/1, 17/1, 2/1, 1/1) which simplifies to:
±34, ±17, ±17/2, ±2, ±1, and ±1/2
To know if any of these are roots, plug them in and see if the result is zero.
Example:
f(34) = 2(34)4 - 8(34)3 + 5(34)2 -3(34) + 34 = 2363952 so, since this is not zero, 34 is not a root.
Keep trying the others to see if any are roots.
f(-34) = 2(-34)4 - 8(-34)3 + 5(-34)2 -3(-34) + 34 = 2993020 so, since this is not zero, -34 is not a root.
Keep checking all the others
