Using the Rational Root Theorem, state all the possible rational roots for the function. Then, solve the function using synthetic division to check and find the actual roots of the function. f(x)=2x^2+3x+1
The possible rational roots for the function f(x) = 2x^2 + 3x + 1 can be found using the Rational Root Theorem. According to the theorem, the possible rational roots are all the factors of the constant term (1) divided by all the factors of the leading coefficient (2).
Factors of 1: ±1
Factors of 2: ±1, ±2
Possible rational roots: ±1/1, ±1/2
Now, let's use synthetic division to check these possible rational roots and find the actual roots of the function.
For x = -1:
-1 | 2 3 1
-2 -1
_____________
2 1 0
Since the remainder is 0, x = -1 is a root of the function.
For x = 1:
1 | 2 3 1
2 5
_____________
2 5 6
Since the remainder is not 0, x = 1 is not a root of the function.
For x = -1/2:
-1/2 | 2 3 1
-1/2 -1/4
_____________
2 2 0
Since the remainder is 0, x = -1/2 is a root of the function.
Therefore, the rational roots of f(x) = 2x^2 + 3x + 1 are x = -1 and x = -1/2.