Question

Consider the following equation: \[ x^3 - 10x^2 + 31x - 30 = 0 \] Which of the following is/are the root(s) of the above equation?

• A. 1
• B. 2
• C. 3
• D. 4

Hint:

When solving cubic equations, start by testing possible integer roots using the Rational Root Theorem, and then proceed with polynomial division to simplify the equation.

Answer 1

The Correct Option is B, C

We are given the cubic equation \( x^3 - 10x^2 + 31x - 30 = 0 \). To find the roots, we can start by checking possible integer roots using the Rational Root Theorem. The possible rational roots are the factors of the constant term (30) divided by the factors of the leading coefficient (1), which are: \( \pm 1, \pm 2, \pm 3, \pm 5, \pm 6, \pm 10, \pm 15, \pm 30 \). Step 1: Check \( x = 2 \)
Substitute \( x = 2 \) into the equation: \[ 2^3 - 10(2)^2 + 31(2) - 30 = 8 - 40 + 62 - 30 = 0 \] Thus, \( x = 2 \) is a root. Step 2: Check \( x = 3 \)
Substitute \( x = 3 \) into the equation: \[ 3^3 - 10(3)^2 + 31(3) - 30 = 27 - 90 + 93 - 30 = 0 \] Thus, \( x = 3 \) is also a root. Final Answer: \[ \boxed{\text{(B) 2, (C) 3}} \]