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The problem presented in the image is about polynomial division and asks whether the product of the roots of \( p(x) = x^3 - 5x^2 + 6x \) is equal to 6.

### Step-by-Step Solution:

1. **Given Polynomials**:
   - \( p(x) = x^3 - 5x^2 + 6x \)
   - \( d(x) = x - 3 \)

2. **Factoring \( p(x) \)**:
   To find the roots of \( p(x) \), we can factor it. Start by factoring out the common factor \( x \):
   \[
   p(x) = x(x^2 - 5x + 6)
   \]
   Now, factor the quadratic \( x^2 - 5x + 6 \):
   \[
   x^2 - 5x + 6 = (x - 2)(x - 3)
   \]
   Thus, we have the fully factored form:
   \[
   p(x) = x(x - 2)(x - 3)
   \]

3. **Roots of \( p(x) \)**:
   The roots of \( p(x) = 0 \) are:
   \[
   x = 0, \quad x = 2, \quad x = 3
   \]

4. **Product of the Roots**:
   According to Vieta's formulas, the product of the roots of a cubic equation \( ax^3 + bx^2 + cx + d = 0 \) is given by:
   \[
   \text{Product of the roots} = \frac{-d}{a}
   \]
   However, since the cubic has been factored as \( x(x - 2)(x - 3) \), we can directly multiply the roots:
   \[
   0 \times 2 \times 3 = 0
   \]

### Conclusion:
The product of the roots of \( p(x) \) is 0, not 6. Therefore, the statement in the question is **False**.

Would you like more details or have any questions?

Here are 5 related questions for further practice:
1. How do you factor cubic polynomials in general?
2. What is the relationship between the roots and coefficients of a polynomial?
3. Can you find the sum of the roots of the polynomial \( p(x) \)?
4. What is synthetic division, and how does it help in dividing polynomials?
5. How does the remainder theorem apply to polynomial division?

**Tip**: Always verify the roots by substituting them back into the original polynomial.

Considere os polinômios p(x) = x^3 - 5x^2 + 6x e d(x) = x - 3, e seja q(x) o quociente da divisão de p(x) por d(x), cujo resto é representado por r(x). Nesse caso, é correto afirmar que o produto das raízes de p(x) é igual a 6.

Learn how to factor and find the roots of the polynomial p(x) = x^3 - 5x^2 + 6x, and apply Vieta's formulas to determine the product of the roots. The solution also explores polynomial division and remainder theorem.

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