Find the zeros of this function: 2x^{3}-x^{2}-6x+3

To find the zeros of the function \( f(x) = 2x^{3} - x^{2} - 6x + 3 \), we need to solve the equation \( f(x) = 0 \).

So, we need to solve:

\[
2x^{3} - x^{2} - 6x + 3 = 0
\]

### Step 1: Check for Rational Roots
We can start by checking for rational roots using the Rational Root Theorem. According to the theorem, any rational root, \( p/q \), must be a factor of the constant term (3) divided by a factor of the leading coefficient (2).

Factors of 3: \( \pm 1, \pm 3 \)  
Factors of 2: \( \pm 1, \pm 2 \)

Possible rational roots: \( \pm 1, \pm \frac{1}{2}, \pm 3, \pm \frac{3}{2} \)

### Step 2: Test the Possible Rational Roots
Let's test these values by substituting them into the function:

1. \( f(1) = 2(1)^3 - (1)^2 - 6(1) + 3 = 2 - 1 - 6 + 3 = -2 \)
2. \( f(-1) = 2(-1)^3 - (-1)^2 - 6(-1) + 3 = -2 - 1 + 6 + 3 = 6 \)
3. \( f(\frac{1}{2}) = 2\left(\frac{1}{2}\right)^3 - \left(\frac{1}{2}\right)^2 - 6\left(\frac{1}{2}\right) + 3 = 2\left(\frac{1}{8}\right) - \frac{1}{4} - 3 + 3 = \frac{1}{4} - \frac{1}{4} = 0 \)

So, \( x = \frac{1}{2} \) is a root of the equation.

### Step 3: Factor the Polynomial
Since \( x = \frac{1}{2} \) is a root, we can factor the polynomial using synthetic division or polynomial division.

Dividing \( 2x^3 - x^2 - 6x + 3 \) by \( x - \frac{1}{2} \), the quotient is \( 2x^2 - 2x - 6 \).

### Step 4: Solve the Remaining Quadratic
We now need to solve the quadratic equation \( 2x^2 - 2x - 6 = 0 \).

Divide the entire equation by 2:

\[
x^2 - x - 3 = 0
\]

We can solve this quadratic equation using the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

where \( a = 1 \), \( b = -1 \), and \( c = -3 \):

\[
x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(1)(-3)}}{2(1)}
\]
\[
x = \frac{1 \pm \sqrt{1 + 12}}{2} = \frac{1 \pm \sqrt{13}}{2}
\]

### Step 5: Conclusion
The zeros of the function are:

\[
x = \frac{1}{2}, \quad x = \frac{1 + \sqrt{13}}{2}, \quad x = \frac{1 - \sqrt{13}}{2}
\]

Would you like more details or have any questions about this process?

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Here are 5 related questions:
1. How do you use the Rational Root Theorem to find potential roots?
2. What is the process for performing synthetic division with polynomials?
3. How can you verify that a value is a root of a polynomial?
4. How do you derive the quadratic formula from the general quadratic equation?
5. What are the applications of finding the zeros of a polynomial in real-world problems?

**Tip:** When solving higher-degree polynomials, always start by checking for simple rational roots before moving on to more complex methods.

Discover how to find the zeros of the polynomial 2x^3 - x^2 - 6x + 3 step-by-step. Learn about polynomial roots, rational root theorem, and quadratic formula application. Suitable for high school students in Grades 10-12.

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