3x^3-11x^2+16x-12=0 when one of the roots is x=2

To solve the cubic equation \( 3x^3 - 11x^2 + 16x - 12 = 0 \) given that one of the roots is \( x = 2 \), we can apply **synthetic division** to factor the cubic polynomial and reduce it to a quadratic. Then, we'll solve the quadratic for the remaining two roots.

### Step 1: Perform synthetic division

We'll divide \( 3x^3 - 11x^2 + 16x - 12 \) by \( (x - 2) \) using synthetic division.

The coefficients of the cubic polynomial are: \( 3, -11, 16, -12 \).

Performing synthetic division:

\[
\begin{array}{r|rrrr}
2 & 3 & -11 & 16 & -12 \\
  &   & 6  & -10 & 12 \\
\hline
  & 3 & -5  & 6   & 0 \\
\end{array}
\]

The quotient is \( 3x^2 - 5x + 6 \), and the remainder is 0, which confirms that \( x = 2 \) is indeed a root.

### Step 2: Solve the quadratic \( 3x^2 - 5x + 6 = 0 \)

Now, we solve the quadratic \( 3x^2 - 5x + 6 = 0 \) using the quadratic formula:

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

For \( 3x^2 - 5x + 6 \), we have:
- \( a = 3 \)
- \( b = -5 \)
- \( c = 6 \)

Substitute the values into the quadratic formula:

\[
x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(3)(6)}}{2(3)}
\]
\[
x = \frac{5 \pm \sqrt{25 - 72}}{6}
\]
\[
x = \frac{5 \pm \sqrt{-47}}{6}
\]
\[
x = \frac{5 \pm i\sqrt{47}}{6}
\]

So, the remaining roots are complex numbers:
\[
x = \frac{5 + i\sqrt{47}}{6} \quad \text{and} \quad x = \frac{5 - i\sqrt{47}}{6}
\]

### Final Answer:
The roots of the equation \( 3x^3 - 11x^2 + 16x - 12 = 0 \) are:
- \( x = 2 \) (real root)
- \( x = \frac{5 + i\sqrt{47}}{6} \) (complex root)
- \( x = \frac{5 - i\sqrt{47}}{6} \) (complex root)

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Would you like more details on synthetic division or the quadratic formula? Feel free to ask!

Here are five related questions for deeper understanding:
1. How does synthetic division work for dividing polynomials?
2. Can a cubic equation have three real roots, and under what conditions?
3. How do complex conjugates appear as roots of polynomials?
4. What does the discriminant \( b^2 - 4ac \) tell us about the nature of the roots?
5. How can we verify if a root is correct using substitution?

**Tip:** For any cubic polynomial, if you know one root, you can always reduce it to a quadratic and solve for the other roots using basic methods like the quadratic formula!

3x^3 - 11x^2 + 16x - 12 = 0 when one of the roots is x = 2

Discover how to solve the cubic equation 3x^3 - 11x^2 + 16x - 12 when one root is known. This step-by-step guide covers synthetic division, solving the resulting quadratic, and finding complex roots. Ideal for students in Grades 10-12.

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