factorise x^3+6x^2+11x-6 =0 using synthetic division

To factorize \(x^3 + 6x^2 + 11x - 6 = 0\) using synthetic division, follow these steps:

### Step 1: Find a possible root using the Rational Root Theorem.
The possible rational roots are factors of the constant term (-6) divided by factors of the leading coefficient (1). So the possible roots are:
\[
\pm 1, \pm 2, \pm 3, \pm 6
\]

### Step 2: Test for roots using synthetic division.
Let's start by testing \(x = 1\).

Set up synthetic division:

\[
\begin{array}{r|rrrr}
1 & 1 & 6 & 11 & -6 \\
  &   & 1 & 7  & 18 \\
\hline
  & 1 & 7 & 18 & 12
\end{array}
\]

Since the remainder is not zero, \(x = 1\) is not a root. Now let's try \(x = -1\).

### Step 3: Try \(x = -1\) using synthetic division.

\[
\begin{array}{r|rrrr}
-1 & 1 & 6 & 11 & -6 \\
   &   & -1 & -5 & -6 \\
\hline
   & 1 & 5  & 6  & 0
\end{array}
\]

The remainder is zero, which means \(x = -1\) is a root. Therefore, \(x^3 + 6x^2 + 11x - 6\) can be factorized as:
\[
(x + 1)(x^2 + 5x + 6)
\]

### Step 4: Factorize the quadratic \(x^2 + 5x + 6\).
Now factor the quadratic:
\[
x^2 + 5x + 6 = (x + 2)(x + 3)
\]

### Step 5: Write the fully factorized form.
Thus, the full factorization of \(x^3 + 6x^2 + 11x - 6 = 0\) is:
\[
(x + 1)(x + 2)(x + 3) = 0
\]

The roots of the equation are \(x = -1\), \(x = -2\), and \(x = -3\).

Would you like further details on any step or have any questions?

### Related Questions:
1. How is the Rational Root Theorem applied in synthetic division?
2. Can synthetic division be used for higher-degree polynomials?
3. How do we factor quadratics like \(x^2 + 5x + 6\)?
4. What are the common mistakes in synthetic division?
5. How does synthetic division compare to long division?

**Tip:** Always test simpler potential roots first, like \( \pm 1 \), to minimize calculations.

Factorise x^3 + 6x^2 + 11x - 6 = 0 using synthetic division.

Learn how to factorize the cubic polynomial x^3 + 6x^2 + 11x - 6 using synthetic division. This detailed solution explains the application of the Rational Root Theorem and synthetic division to find the roots and factor the polynomial. Suitable for students in Grades 10-12.

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