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

Key Concepts Needed

1. Factorisation of Polynomials
   • Break a big polynomial into simpler factors like $(x-1)$, $(x+1)$, etc.
2. Multiplicity vs. Distinct Roots
   • If a factor repeats (e.g., $(x-1)^2$), it is the same root counted more than once. When we are asked for distinct roots, we count it only once.
3. Monotonic Functions
   • If the derivative of a polynomial is always positive, the polynomial is strictly increasing → it can cross the $x$-axis at most once, giving exactly one real root.

Logical Chain to Solve

1. Expand or Spot a Factor: Notice $x^3 - x^2 - x + 1 = (x-1)^2(x+1)$
2. Write Equation as One Polynomial: Expand the given expression so we can factor it cleanly.
3. Synthetic (or Long) Division: Once we guess a small root like $x=1$, divide the polynomial by $(x-1)$ to peel off that factor.
4. Repeat Until No Obvious Factors: Keep dividing whenever you find another easy root ($x=1$ again, then $x=-1$).
5. Left-over Quintic: After removing the easy factors, you are left with a fifth-degree part $x^5 + 3x - 1$.
6. Use the Derivative: Compute its derivative $5x^4 + 3$ → always positive → function is strictly increasing → one real root.
7. Count Distinct Roots: Combine: $(x-1)^2$ gives one root, $(x+1)$ gives one more, quintic gives one. Total 3 distinct real roots.

Imagine a Giant Lego Tower

1. Big Expression = Tower – The long equation is like a very tall Lego tower built out of smaller blocks (factors).
2. Find Small Blocks – If we break the tower into blocks, each block makes solving easier.
3. Blocks Give Heights (Roots) – Every time a block becomes zero, the whole tower touches the ground. Those touching points are the roots.
4. Some Blocks Repeat – A repeated block touches the ground in the same place twice, but we count the touching point only once.
5. One Special Block Grows Only Up – One big block keeps going up without turning back. Because it always rises, it can cross the ground only once.
6. Total Touch-Points – Add up all different places where it touches the ground. Here, we get three different spots.

So, the tower touches the ground at 3 distinct places → 3 real roots.

Step-by-Step Solution

1. Rewrite the Cubic Factor

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

2. Expand Whole Equation to One Polynomial

   $$\begin{aligned} &x^5\bigl(x^3 - x^2 - x + 1\bigr) + x\bigl(3x^3 - 4x^2 - 2x + 4\bigr) - 1 \\ &= x^8 - x^7 - x^6 + x^5 + 3x^4 - 4x^3 - 2x^2 + 4x - 1. \end{aligned}$$

3. Call This $P(x)$ and Test Easy Integers

   • $P(1)=0$ ⇒ $(x-1)$ is a factor.
   • Perform synthetic division twice to reveal multiplicity.

4. Synthetic Division Results

   After dividing by $(x-1)$ twice and by $(x+1)$ once, we get

   $P(x) = (x-1)^2 (x+1) \bigl(x^5 + 3x -1\bigr).$

5. Multiplicity & Distinct Roots

   • $(x-1)^2 = 0$ ⇒ root $x=1$ (multiplicity 2, but 1 distinct root).
   • $(x+1)=0$ ⇒ root $x=-1$ (multiplicity 1).

6. Analyse the Quintic

   • Let $f(x)=x^5+3x-1$.
   • Derivative $f'(x)=5x^4+3>0$ for all real $x$.
   • Therefore, $f(x)$ is strictly increasing ⇒ crosses the $x$-axis exactly once ⇒ one real root.

7. Total Distinct Real Roots

   $1\,(\text{from }x=1) + 1\,(x=-1) + 1\,(\text{quintic}) = 3.$

[ \boxed{\text{Number of distinct real roots} = 3} ]