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. Show your work.
f(x) = 2x^2+3x+1
A. Possible roots: ±1, ±2, ±3
Actual roots: x = 1 or x = -2
B. Possible roots: ±2, ±1
Actual roots: x = 1 or x = 2
C. Possible roots: ±1, ±1/3
Actual roots: x = -1 or x = 1/3
D. Possible roots: ±1/2, ±1
Actual roots: x = -1 or x = -1/2
First, let's determine all the possible rational roots for the function using the Rational Root Theorem:
The Rational Root Theorem states that for a polynomial function such as:
\[ f(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0 \]
if \( \frac{p}{q} \) is a rational root, where \( p \) is a factor of the constant term \( a_0 \) and \( q \) is a factor of the leading coefficient \( a_n \), then \( p \) must divide \( a_0 \) and \( q \) must divide \( a_n \).
For the function \( f(x) = 2x^2 + 3x + 1 \), the constant term \( a_0 = 1 \) and the leading coefficient \( a_n = 2 \).
The factors of 1 (constant term) are: ±1
The factors of 2 (leading coefficient) are: ±1, ±2
So, the possible rational roots are: \( \pm \frac{1}{1}, \pm \frac{1}{2} \), which simplify to: ±1, ±1/2
Therefore, possible roots are: ±1/2, ±1
This eliminates options A and B.
Now let's use synthetic division to find the actual roots:
For root -1/2:
```
________________
-1/2 | 2 3 1
| -1 1
-------------
2 2 2
```
Since the remainder is not zero, -1/2 is not a root.
For root 1/2:
```
_______________
1/2 | 2 3 1
| 1 2
------------
2 4 3
```
Since the remainder is not zero, 1/2 is not a root.
For root -1:
```
____________
-1 | 2 3 1
| -2 -1
-----------
2 1 0
```
Since the remainder is zero, -1 is a root.
The quotient we have now is 2x + 1, which we can set equal to zero to find the other root.
Solve for x:
\[ 2x + 1 = 0 \]
\[ 2x = -1 \]
\[ x = -\frac{1}{2} \]
Therefore, the actual roots are x = -1 or x = -1/2.
Now we can confidently choose the correct option:
D. Possible roots: ±1/2, ±1
Actual roots: x = -1 or x = -1/2