Question

If $x^2 - 5x + 6 = 0$, what is the value of $x^3 - 3x^2 + 2x$? 
 

• A. 6
• B. 12
• C. 18
• D. 24

Hint:

For polynomial evaluation with given roots, substitute each root directly to find possible values, especially when a single answer is expected.

Answer 1

The Correct Option is A

- Step 1: Solve the quadratic: $x^2 - 5x + 6 = 0$ factors as $(x - 2)(x - 3) = 0$, so $x = 2$ or $x = 3$.
- Step 2: Compute $x^3 - 3x^2 + 2x$ for each root.
- Step 3: For $x = 2$: $2^3 - 3 \times 2^2 + 2 \times 2 = 8 - 12 + 4 = 0$.
- Step 4: For $x = 3$: $3^3 - 3 \times 3^2 + 2 \times 3 = 27 - 27 + 6 = 6$.
- Step 5: Since the question asks for a single value and options suggest one root, test options. Option (a) 6 matches $x = 3$.
- Step 6: Alternatively, use polynomial division to express $x^3 - 3x^2 + 2x$ in terms of the quadratic, but direct substitution is faster.