The sum of 162th power of the roots of the equation x³ - 2x² + 2x – 1 = 0 is

Q: Why can't we just use Vieta's formulas to find the sum of the powers?

Vieta's formulas are excellent for finding the sum of roots, the sum of roots taken two at a time, and the product of roots. However, they do not directly give the sum of the 162nd powers. While Newton's Sums could be used, they would be computationally tedious for such a high power. Factoring the polynomial is the elegant shortcut.

Q: How did you know to multiply the quadratic by $(x+1)$?

This is a classic JEE pattern recognition skill! Whenever you see the quadratic $x^2 - x + 1 = 0$, it is the cyclotomic polynomial for the 6th roots of unity. Multiplying by $(x+1)$ completes the sum of cubes identity $x^3 + 1^3 = 0$. It is a powerful tool to keep in your arsenal for complex number problems.

Q: Wait, why is $x^3 = -1$ and not $1$?

Great catch! The equation is $x^2 - x + 1 = 0$. Multiplying by $(x+1)$ gives $(x+1)(x^2 - x + 1) = x^3 + 1$. Setting this to zero gives $x^3 + 1 = 0$, which means $x^3 = -1$. Always double-check the signs when applying algebraic identities!

Question

The sum of $162^{th}$ power of the roots of the equation $x^3 - 2x^2 + 2x – 1 = 0$ is

Sigma Percentile · Accepted Answer

### The Mystery of the 162nd Power

Imagine you are standing before a formidable, towering mountain of a problem. You are asked to find the sum of the 162nd power of the roots of the equation $x^3 - 2x^2 + 2x - 1 = 0$.

Your first instinct might be to panic. How could anyone possibly calculate the 162nd power of three roots?

But here is the secret of JEE Advanced: the problem is not a test of your endurance; it is a test of your vision. There is a hidden symmetry waiting to be uncovered.

### The Art of Factoring

Let us look at the equation again: $x^3 - 2x^2 + 2x - 1 = 0$. If we stare at it long enough, we see a pattern.

Let us group the terms: $(x^3 - 1) - (2x^2 - 2x) = 0$.

Now, the magic happens. We know the identity $a^3 - b^3 = (a-b)(a^2 + ab + b^2)$. Applying this to the first group, we get $(x-1)(x^2 + x + 1)$.

For the second group, we factor out $2x$, giving us $-2x(x-1)$. Suddenly, the path clears.

We have a common factor of $(x-1)$! Factoring it out, we get $(x-1)(x^2 + x + 1 - 2x) = 0$, which simplifies beautifully to:

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

### The Beauty of the Complex Plane

We have found our first root: $\alpha = 1$. But what about the other two, $\beta$ and $\gamma$?

They satisfy $x^2 - x + 1 = 0$. These are complex roots. Instead of solving for them using the quadratic formula, let us use a clever trick.

Multiply this quadratic by $(x+1)$. This gives us $(x+1)(x^2 - x + 1) = 0$, which is the identity for $x^3 + 1 = 0$.

Thus, for these roots, $x^3 = -1$. This is a profound realization. These roots lie on the unit circle in the complex plane, and their cubes are $-1$.

### The Final Tally

Now, we return to our goal: $\alpha^{162} + \beta^{162} + \gamma^{162}$. Since $\alpha = 1$, $\alpha^{162} = 1$.

For $\beta$ and $\gamma$, we use the property $x^3 = -1$. We can write $x^{162}$ as $(x^3)^{54}$.

Substituting $x^3 = -1$, we get $(-1)^{54}$. Since 54 is an even number, this becomes $1$.

Therefore, $\beta^{162} = 1$ and $\gamma^{162} = 1$. Adding them all together:

$$1 + 1 + 1 = 3$$

We have conquered the mountain, not by brute force, but by the elegance of algebra. The final answer is **3**.

Animated Solution for Mathematics - Complex Numbers: The sum of $16 2^{t h}$ power of the roots of the equation $x^{3} - 2 x^{2} + 2 x -1 = 0$ is

Enter Numerical Value:

Visualized Solution

Analyzing the Cubic Equation

Grouping the Terms

Applying Algebraic Identities

Factoring Out Common Terms

Final Factored Form

Identifying the First Root $α$

Analyzing the Quadratic Factor

A Clever Algebraic Trick

Visualizing the Complex Roots

Setting Up the Final Calculation

Calculating the Powers

Final Summation

The Sigma Insight: Cube Roots and nth Roots of Unity

The Mystery of the 162nd Power

The Art of Factoring

The Beauty of the Complex Plane

The Final Tally

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Animated Solution for Mathematics - Complex Numbers: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The sum of 162th power of the roots of the equation x3−2x2+2x–1=0 is