Using the Rational Root Theorem, state all the possible rational roots for the function.show all your work

Question

Then, solve the function using synthetic division to check and find the actual roots of the function.

f(x)=2x2+3x+1
(1 point)
Responses

Possible roots: ±1, ±2, ±3
Actual roots: x = 1 or x=−2
Possible roots: ±1, ±2, ±3
Actual roots: x = 1 or x=−2

Possible roots: ±1, ±13
Actual roots: x = −1 or x=13
Possible roots: ±1, ±13
Actual roots: x = −1 or x=13

Possible roots: ±2, ±1
Actual roots: x = 1 or x=2
Possible roots: ±2, ±1
Actual roots: x = 1 or x=2

Possible roots: ±12, ±1
Actual roots: x =−1 or x=−12

Answer 1

Possible roots: ±1, ±2

To find the actual roots, we can use synthetic division:

1 | 2 3 1
| 2 5
--------------
2 5 6

Since the remainder is not 0, 1 is not a root.

-1 | 2 3 1
| -2 -1
--------------
2 1 0

Since the remainder is 0, -1 is a root.

2 | 2 3 1
| 4 14
--------------
2 7 15

Since the remainder is not 0, 2 is not a root.

Therefore, the actual roots of the function are x = -1 and x = 2.