Question

Roots of the equation \(x^3 - 23x^2 + 142x - 120 = 0\) are

• A. 1, 10, 12
• B. -1, 10, 12
• C. 1, -10, 12
• D. -1, -10, -12

Hint:

For multiple-choice questions on polynomial roots, always use Vieta's formulas to verify the sum and product of the options.
This is much faster than performing synthetic division or factoring the cubic equation.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This question requires finding the three roots of a cubic polynomial equation.
This can be solved by testing candidates using the rational root theorem, factoring the polynomial, or using Vieta's formulas for cubic equations.

Step 2: Detailed Explanation:
Let us use Vieta's formulas to evaluate the roots. For any cubic equation of the form:
\[ ax^3 + bx^2 + cx + d = 0 \] with roots \( \alpha \), \( \beta \), and \( \gamma \), the formulas state:
1. Sum of the roots:
\[ \alpha + \beta + \gamma = -\frac{b}{a} \] 2. Product of the roots taken two at a time:
\[ \alpha\beta + \beta\gamma + \alpha\gamma = \frac{c}{a} \] 3. Product of the roots:
\[ \alpha\beta\gamma = -\frac{d}{a} \] Let us extract the coefficients from our cubic equation:
\[ x^3 - 23x^2 + 142x - 120 = 0 \] Here, \( a = 1 \), \( b = -23 \), \( c = 142 \), and \( d = -120 \).
- The sum of the roots must be:
\[ \alpha + \beta + \gamma = -\frac{-23}{1} = 23 \] - The product of the roots must be:
\[ \alpha\beta\gamma = -\frac{-120}{1} = 120 \] Let us test the given options against these two simple conditions:
- Option (A): \( 1, 10, 12 \)
- Sum of roots: \( 1 + 10 + 12 = 23 \) (Matches!)
- Product of roots: \( 1 \times 10 \times 12 = 120 \) (Matches!)
Since both conditions are satisfied, these must be the correct roots.
- Option (B): \( -1, 10, 12 \)
- Sum of roots: \( -1 + 10 + 12 = 21 \neq 23 \). (Incorrect)
- Option (C): \( 1, -10, 12 \)
- Sum of roots: \( 1 - 10 + 12 = 3 \neq 23 \). (Incorrect)
- Option (D): \( -1, -10, -12 \)
- Sum of roots: \( -1 - 10 - 12 = -23 \neq 23 \). (Incorrect)
Therefore, only the roots in Option (A) satisfy the cubic polynomial equation.

Step 3: Final Answer:
The roots of the equation are \( 1, 10, 12 \).
Therefore, the correct choice is Option (A).