Question 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)=2x2+3x+1 (1 point) Responses 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: ±1, ±2, ±3 Actual roots: x = 1 or x=−2 Possible roots: ±1, ±2, ±3 Actual roots: x = 1 or x=−2 Possible roots: ±12, ±1 Actual roots: x =−1 or x=−12
To use the Rational Root Theorem on the function f(x) = 2x^2 + 3x + 1, we need to look at the factors of the constant term and the leading coefficient.
The constant term is 1, with factors ±1.
The leading coefficient is 2, with factors ±1, ±2.
According to the Rational Root Theorem, the possible rational roots of the function f(x) are the ratios of the factors of the constant term to the factors of the leading coefficient. So, they are:
Possible rational roots: ±1/1, ±1/2
Simplified, these are: ±1, ±1/2
Now, we can check each possible root using synthetic division to find the actual roots.
Let's try 1:
```
1 | 2 3 1
| 2 5
-----------------
2 5 6
```
Since the remainder is not zero (it's 6), 1 is not a root.
Let's try -1:
```
-1 | 2 3 1
| -2 -1
-----------------
2 1 0
```
The remainder is zero, which means that x = -1 is an actual root of the function.
Finally, let's check 1/2 and -1/2:
Trying 1/2 is unnecessary as the coefficients are integers; if -1 is a root so would be -2 which gives a remainder of 12 at synthetic division, and not zero.
Trying -1/2:
```
-0.5 | 2 3 1
| -1 -1
----------------
2 2 0
```
The remainder is zero again, which means that x = -1/2 is also an actual root of the function.
In conclusion,
Possible roots: ±1, ±1/2
Actual roots: x = -1 or x = -1/2
The correct response is that the possible roots are ±1, ±1/2, and the actual roots of the function are x = -1 and x = -1/2.