The number of distinct real roots of x⁴ - 4x³ + 12x² + x - 1 = 0 is

• A) 2
• B) 3
• C) 4
• D) 5

To determine the number of distinct real roots of the polynomial equation \(x^4 - 4x^3 + 12x^2 + x - 1 = 0\), we can use calculus and the concept of the derivative.

Step 1: Analyze the Polynomial

The given polynomial is a quartic function, which means it can have up to 4 real roots. To find the number of distinct real roots, we will first look at its derivative.

Finding the Derivative

The derivative of the polynomial is:

f'(x) = 4x^3 - 12x^2 + 24x + 1

Step 2: Critical Points

Next, we need to find the critical points by setting the derivative equal to zero:

4x^3 - 12x^2 + 24x + 1 = 0

Finding the roots of this cubic equation will help us determine the behavior of the original polynomial.

Using the Discriminant

The discriminant of a cubic equation can indicate the number of real roots. If the discriminant is positive, there are three distinct real roots; if it is zero, there is one real root; and if it is negative, there is one real root and two complex roots.

Step 3: Evaluating the Polynomial

We can also evaluate the original polynomial at various points to check for sign changes, which indicate the presence of roots:

• f(-2) = 25
• f(-1) = 15
• f(0) = -1
• f(1) = 9
• f(2) = 37

From these evaluations, we see that there is a sign change between \(f(0)\) and \(f(1)\), indicating at least one root in that interval.

Conclusion on Roots

By analyzing the critical points and evaluating the polynomial, we can conclude that the polynomial has 2 distinct real roots. Therefore, the answer is:

A) 2