Question

The number of distinct real roots of the equation $3x^4 + 4x^3 - 12x^2 + 4 = 0$ is _________.

Hint:

Drawing a rough sketch of the polynomial using its turning points and endpoints is the most reliable way to count real roots.

Answer 1

Correct Answer: 4

Step 1: Understanding the Concept:
To find the number of real roots of a polynomial equation, we examine its derivative to identify intervals of monotonicity and look for sign changes in the function values at local extrema.
Step 2: Detailed Explanation:
Let $f(x) = 3x^4 + 4x^3 - 12x^2 + 4$.
Differentiating: $f'(x) = 12x^3 + 12x^2 - 24x = 12x(x^2 + x - 2) = 12x(x + 2)(x - 1)$.
The critical points are $x = -2, x = 0,$ and $x = 1$.
Calculate function values at these points:
$f(-2) = 3(16) + 4(-8) - 12(4) + 4 = 48 - 32 - 48 + 4 = -28$.
$f(0) = 4$.
$f(1) = 3(1) + 4(1) - 12(1) + 4 = 7 - 12 + 4 = -1$.
Behavior at infinity:
$\lim_{x \to \infty} f(x) = \infty$ and $\lim_{x \to -\infty} f(x) = \infty$.
Analyze sign changes:
1. In $(-\infty, -2)$: $f(-\infty) = \infty$ to $f(-2) = -28 \implies 1$ root.
2. In $(-2, 0)$: $f(-2) = -28$ to $f(0) = 4 \implies 1$ root.
3. In $(0, 1)$: $f(0) = 4$ to $f(1) = -1 \implies 1$ root.
4. In $(1, \infty)$: $f(1) = -1$ to $f(\infty) = \infty \implies 1$ root.
Total distinct real roots $= 1 + 1 + 1 + 1 = 4$.
Step 3: Final Answer:
The number of distinct real roots is 4.