Question

The number of distinct real roots of \(x^4 – 4x + 1 = 0\) is :

• A. 4
• B. 2
• C. 1
• D. 0

Answer 1

The Correct Option is B

 To find the number of distinct real roots of the polynomial equation \(x^4 - 4x + 1 = 0\), we can analyze the function \(f(x) = x^4 - 4x + 1\).

We will use calculus to determine the number of distinct real roots by examining the critical points and the behavior of the function:

1. First, find the derivative of \(f(x)\):
2. \(f'(x) = \frac{d}{dx}(x^4 - 4x + 1) = 4x^3 - 4\)
3. Find the critical points by solving \(f'(x) = 0\):
4. \(4x^3 - 4 = 0 \implies 4(x^3 - 1) = 0 \implies x^3 - 1 = 0\)
5. Solving \(x^3 - 1 = 0\), we find \(x = 1\).
6. Analyze the second derivative to determine concavity and the nature of the critical point:
7. \(f''(x) = \frac{d}{dx}(4x^3 - 4) = 12x^2\)
8. Evaluate the second derivative at the critical point \(x = 1\):
9. \('(1) = 12(1)^2 = 12\), which is positive, indicating that \(f(x)\) has a local minimum at \(x = 1\\)

Next, examine the behavior of the function:

• As \(x \to \pm \infty\), \(f(x) = x^4 - 4x + 1\) tends towards \(+\infty\) because the degree of the polynomial is even.
• At \(x = 1\), \(f(1) = 1^4 - 4(1) + 1 = -2\). This indicates the point is below the x-axis, confirming a minimum.

To find the roots, we should also consider other aspects, such as symmetry:

• \(f(x)\) is an even function as \(f(x) \ne f(-x)\). Thus, symmetry about the y-axis is not applicable. However, symmetry around the origin can imply additional roots.

The transition of \(f(x)\) from \(+\infty\) to \(-2\) and then again back to \(+\infty\) as \(x\) crosses \(x = 1\) implies that there are exactly two real roots, confirmed by checking values and changes in signs around critical points.

Therefore, the number of distinct real roots of \(x^4 - 4x + 1 = 0\) is 2.