The number of distinct real roots of the equation x⁵(x³ - x² - x + 1) + x(3x³ - 4x² - 2x + 4) - 1 =

Q: Why can't we just expand the whole equation and solve it like a normal polynomial?

That is a classic trap! Expanding a degree-8 polynomial is a nightmare and will likely lead to calculation errors. In JEE Advanced, these problems are designed to test your ability to spot structure and symmetry. Always look for common factors before resorting to brute force.

Q: How do I know $x=1$ and $x=-1$ are roots of the quartic $3x^4 - 4x^3 - 2x^2 + 4x - 1$?

Great question! When you see a polynomial, always check for 'easy' roots first. For $x=1$, the sum of coefficients is $3 - 4 - 2 + 4 - 1 = 0$, which confirms $x=1$ is a root. For $x=-1$, substituting gives $3 + 4 - 2 - 4 - 1 = 0$. Once you find these, you can divide the polynomial by $(x-1)(x+1) = x^2 - 1$ to find the remaining quadratic factor.

Q: Why does $f'(x) > 0$ guarantee only one root for $f(x) = x^5 + 3x - 1$?

Think of the derivative as the 'slope' of the function. If $f'(x) > 0$ for all $x$, the function is strictly increasing—it is always climbing uphill. A function that is always climbing can cross the x-axis at most once. Combined with the Intermediate Value Theorem, which shows the function changes sign, we can be absolutely certain there is exactly one root.

Question

The number of distinct real roots of the equation $x^5(x^3 - x^2 - x + 1) + x(3x^3 - 4x^2 - 2x + 4) - 1 = 0$ is ______.

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are sitting in the exam hall, and you see this equation: 
$$x^5(x^3 - x^2 - x + 1) + x(3x^3 - 4x^2 - 2x + 4) - 1 = 0$$

Your first instinct might be to expand everything, but take a deep breath. In the world of JEE Advanced, brute force is rarely the intended path. This problem is a test of your ability to see the hidden architecture beneath the surface.

### Unveiling the Hidden Structure

Instead of expanding, let us focus on the first bracket: $x^3 - x^2 - x + 1$. If we group the terms, we get $x^2(x - 1) - 1(x - 1)$, which factors beautifully into $(x^2 - 1)(x - 1)$, or more simply, $(x - 1)^2(x + 1)$.

Now, look at the second part: $x(3x^3 - 4x^2 - 2x + 4) - 1$. Expanding the $x$ gives us $3x^4 - 4x^3 - 2x^2 + 4x - 1$.

Testing $x=1$ and $x=-1$ reveals they are roots of this quartic. By dividing this quartic by $(x^2 - 1)$, we find the remaining factor is $(3x^2 - 4x + 1)$, which further factors into $(3x - 1)(x - 1)$. Thus, the second part is $(x - 1)^2(x + 1)(3x - 1)$.

### The Elegant Cancellation

Now, let us bring the pieces together. Substituting our factored forms back into the original equation, we get:

$$x^5(x - 1)^2(x + 1) + (x - 1)^2(x + 1)(3x - 1) = 0$$

Do you see it? The term $(x - 1)^2(x + 1)$ is common to both! Factoring it out, we are left with:

$$(x - 1)^2(x + 1)(x^5 + 3x - 1) = 0$$

This is the 'Aha!' moment. We have reduced a complex, high-degree equation into a product of simpler factors.

### The Calculus Toolkit

We have two obvious roots from the first two factors: $x = 1$ and $x = -1$. But what about the quintic factor, $f(x) = x^5 + 3x - 1$?

To find its roots, we turn to calculus. Differentiating gives:

$$f'(x) = 5x^4 + 3$$

Since $x^4$ is always non-negative, $f'(x)$ is always positive. This means $f(x)$ is a **strictly increasing function**. A strictly increasing function can cross the x-axis at most once.

Using the Intermediate Value Theorem, we check $f(0) = -1$ and $f(1) = 3$. Since the function changes from negative to positive, there must be a root $\beta$ in the interval $(0, 1)$.

### The Final Tally

We have found three distinct real roots: $x = 1$, $x = -1$, and $x = \beta$.

The beauty of this problem lies not in the calculation, but in the realization that structure always triumphs over complexity. You have successfully navigated the trap and emerged with the correct answer: **3**.

Animated Solution for Mathematics - Differentiation: The number of distinct real roots of the equation $x^{5} (x^{3} - x^{2} - x + 1) + x (3 x^{3} - 4 x^{2} - 2 x + 4) - 1 = 0$ is ______.

Enter Numerical Value:

Visualized Solution

Analyze the Equation

Factorizing the First Term

Expanding the Second Part

Factorizing the Quartic

Combining the Terms

Identifying Obvious Roots

Defining the Function $f (x)$

Analyzing the Derivative

Monotonicity of $f (x)$

Intermediate Value Theorem

Final Count of Distinct Roots

The Sigma Insight: Monotonicity

Analyzing the Setup

Unveiling the Hidden Structure

The Elegant Cancellation

The Calculus Toolkit

The Final Tally

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Animated Solution for Mathematics - Differentiation: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of distinct real roots of the equation x5(x3−x2−x+1)+x(3x3−4x2−2x+4)−1=0 is ______.