Using the Rational Root Theorem, state all the possible rational roots for the function. show all your work
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
To find the possible rational roots of the function f(x) = 2x^2 + 3x + 1, we need to consider the factors of the constant term (1) and the factors of the leading coefficient (2).
The factors of the constant term 1 are ±1.
The factors of the leading coefficient 2 are ±1, ±2.
Therefore, the possible rational roots of the function are ±1 and ±2.
We can now use synthetic division to check these possible roots and find the actual roots of the function:
Synthetic Division with x = 1:
1 | 2 3 1
| 2 5
|_____________
2 5 6
Since the remainder is 6, the root x = 1 is not a solution to the equation.
Synthetic Division with x = -1:
-1 | 2 3 1
-2 -1
|_____________
2 1 0
Since the remainder is 0, the root x = -1 is a solution to the equation.
Synthetic Division with x = 2:
2 | 2 3 1
4 14
|_____________
2 7 15
Since the remainder is not 0, the root x = 2 is not a solution to the equation.
Therefore, the actual roots of the function f(x) = 2x^2 + 3x + 1 are x = -1.