Find the roots of the polynomial (2x − 1)(x

3 + 2x

2 − 5x − 6).

To find the roots of the polynomial \((2x - 1)(x^3 + 2x^2 - 5x - 6)\), we need to set each factor equal to zero and solve for \(x\).

### Step 1: Set each factor equal to zero.

1. **First Factor: \(2x - 1 = 0\)**

   Solve for \(x\):
   \[
   2x = 1 \quad \Rightarrow \quad x = \frac{1}{2}
   \]
   So, \(x = \frac{1}{2}\) is one root.

2. **Second Factor: \(x^3 + 2x^2 - 5x - 6 = 0\)**

   This is a cubic equation, and we need to find its roots.

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

Let's try to find one root by using the Rational Root Theorem, which suggests testing factors of the constant term (-6) over factors of the leading coefficient (1). The possible rational roots are \(\pm 1\), \(\pm 2\), \(\pm 3\), \(\pm 6\).

Testing these:

- \(x = -2\) is a root because:
  \[
  (-2)^3 + 2(-2)^2 - 5(-2) - 6 = -8 + 8 + 10 - 6 = 0
  \]

### Step 3: Factorize the cubic polynomial using \(x = -2\).

Since \(x = -2\) is a root, we can factorize the cubic polynomial by dividing \(x^3 + 2x^2 - 5x - 6\) by \(x + 2\) using synthetic division or long division.

The quotient is \(x^2 - 2x - 3\), so:
\[
x^3 + 2x^2 - 5x - 6 = (x + 2)(x^2 - 2x - 3)
\]

### Step 4: Solve the quadratic equation \(x^2 - 2x - 3 = 0\).

Factorizing the quadratic:
\[
x^2 - 2x - 3 = (x - 3)(x + 1) = 0
\]

So, the roots are \(x = 3\) and \(x = -1\).

### Final Answer: The roots of the polynomial are:
\[
x = \frac{1}{2}, -2, 3, -1
\]

Would you like details or have any questions?

Here are 5 related questions:
1. How do you perform synthetic division on polynomials?
2. Can you explain the Rational Root Theorem in detail?
3. What are the steps to factor a cubic polynomial?
4. How can you check if a root is correct for a polynomial equation?
5. What are the possible number of real roots for a cubic polynomial?

**Tip:** When solving polynomials, always try to factorize first before using more advanced techniques like the quadratic formula or synthetic division.

Learn how to find the roots of the polynomial (2x - 1)(x^3 + 2x^2 - 5x - 6) with a detailed step-by-step solution. This problem covers key algebraic concepts such as polynomials, factorization, and the Rational Root Theorem. Suitable for students in Grades 10-12.

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