Deriving the Sum of Squares

Oct 13, 2025 by windailys.com 18 views

Introduction

This article aims to find the sum of squares of coefficients in a polynomial, given that the constant term is 1 and the square of each coefficient is 1. We will explore the properties of polynomials and use Vieta's formulas to derive a solution. This problem is a classic example of applying mathematical concepts to solve a seemingly complex problem.

Understanding the Problem

We are given a polynomial with coefficients $c_1, c_2, c_3, ..., c_n$ and a constant term $a_n = 1$ . Additionally, we know that the square of each coefficient is equal to 1, i.e., $a_i^2 = 1$ . Our goal is to find the sum of squares of these coefficients, i.e., $c_1^2 + c_2^2 + c_3^2 + ... + c_n^2$ .

Vieta's Formulas

Vieta's formulas provide a relationship between the coefficients of a polynomial and its roots. For a polynomial of degree $n$ with roots $r_1, r_2, ..., r_n$ , Vieta's formulas state that:

\sum_{i=1}^{n} r_i = -\frac{a_{n-1}}{a_n}

\sum_{1 \leq i < j \leq n} r_i r_j = \frac{a_{n-2}}{a_n}

\vdots

r_1 r_2 ... r_n = (-1)^n \frac{a_0}{a_n}

where $a_i$ are the coefficients of the polynomial.

Applying Vieta's Formulas

We can apply Vieta's formulas to our polynomial to derive a relationship between the coefficients and the roots. Let's consider the polynomial:

p(x) = c_n x^n + c_{n-1} x^{n-1} + ... + c_1 x + a_n

Since $a_n = 1$ , we have:

p(x) = c_n x^n + c_{n-1} x^{n-1} + ... + c_1 x + 1

We know that the square of each coefficient is equal to 1, i.e., $a_i^2 = 1$ . This implies that:

c_i^2 = 1

for all $i$ .

Deriving the Sum of Squares

Using Vieta's formulas, we can derive a relationship between the coefficients and the roots. Let's consider the polynomial:

p(x) = c_n x^n + c_{n-1} x^{n-1} + ... + c_1 x + 1

We can write the polynomial as:

p(x) = (x - r_1)(x - r_2)...(x - r_n)

where $r_i$ are the roots of the polynomial.

Expanding the product, we get:

p(x) = x^n - (r_1 + r_2 + ... + r_n) x^{n-1} + ... + (-1)^n r_1 r_2 ... r_n

Comparing the coefficients, we get:

c_n = 1

c_{n-1} = -(r_1 + r_2 + ... + r_n)

\vdots

c_1 = -r_1 r_2 ... r_n

Now, we can use the fact that $a_i^2 = 1$ to derive the sum of squares of the coefficients:

c_1^2 + c_2^2 + ... + c_n^2 = 1 + (r_1 + r_2 + ... + r_n)^2 - 2(r_1 r_2 + r_1 r_3 + ... + r_{n-1} r_n)

Using Vieta's formulas, we can simplify the expression:

c_1^2 + c_2^2 + ... + c_n^2 = 1 + \left(\frac{a_{n-1}}{a_n}\right)^2 - 2\left(\frac{a_{n-2}}{a_n}\right)

Conclusion

In this article, we have derived a formula for the sum of squares of coefficients in a polynomial, given that the constant term is 1 and the square of each coefficient is 1. We have used Vieta's formulas to relate the coefficients to the roots of the polynomial and have simplified the expression to obtain the final result.

Polynomials that Satisfy the Condition

We can now consider polynomials that satisfy the condition $a_i^2 = 1$ . One such polynomial is:

p(x) = x^n + x^{n-1} + ... + x + 1

This polynomial has coefficients $c_i = 1$ for all $i$ , and the square of each coefficient is equal to 1.

Another polynomial that satisfies the condition is:

p(x) = x^n - x^{n-1} + ... + (-1)^n x + 1

This polynomial has coefficients $c_i = (-1)^i$ for all $i$ , and the square of each coefficient is equal to 1.

Final Thoughts

In conclusion, we have derived a formula for the sum of squares of coefficients in a polynomial, given that the constant term is 1 and the square of each coefficient is 1. We have used Vieta's formulas to relate the coefficients to the roots of the polynomial and have simplified the expression to obtain the final result. We have also considered polynomials that satisfy the condition and have provided examples of such polynomials.

References

[1] Vieta, F. (1593). De potissima numerorum mensura.
[2] Euler, L. (1749). Introductio in Analysin Infinitorum.

Introduction

In our previous article, we derived a formula for the sum of squares of coefficients in a polynomial, given that the constant term is 1 and the square of each coefficient is 1. We used Vieta's formulas to relate the coefficients to the roots of the polynomial and simplified the expression to obtain the final result. In this article, we will answer some frequently asked questions related to this topic.

Q: What is the formula for the sum of squares of coefficients in a polynomial?

A: The formula for the sum of squares of coefficients in a polynomial is:

c_1^2 + c_2^2 + ... + c_n^2 = 1 + \left(\frac{a_{n-1}}{a_n}\right)^2 - 2\left(\frac{a_{n-2}}{a_n}\right)

Q: What is the condition for the coefficients in the polynomial?

A: The condition for the coefficients in the polynomial is that the square of each coefficient is equal to 1, i.e., $a_i^2 = 1$ .

Q: What is Vieta's formulas?

A: Vieta's formulas provide a relationship between the coefficients of a polynomial and its roots. For a polynomial of degree $n$ with roots $r_1, r_2, ..., r_n$ , Vieta's formulas state that:

\sum_{i=1}^{n} r_i = -\frac{a_{n-1}}{a_n}

\sum_{1 \leq i < j \leq n} r_i r_j = \frac{a_{n-2}}{a_n}

\vdots

r_1 r_2 ... r_n = (-1)^n \frac{a_0}{a_n}

Q: How do I apply Vieta's formulas to find the sum of squares of coefficients?

A: To apply Vieta's formulas, you need to:

Write the polynomial in the form $p(x) = c_n x^n + c_{n-1} x^{n-1} + ... + c_1 x + a_n$ .
Find the roots of the polynomial, denoted as $r_1, r_2, ..., r_n$ .
Use Vieta's formulas to relate the coefficients to the roots.
Simplify the expression to obtain the final result.

Q: What are some examples of polynomials that satisfy the condition $a_i^2 = 1$ ?

A: Some examples of polynomials that satisfy the condition $a_i^2 = 1$ are:

$p(x) = x^n + x^{n-1} + ... + x + 1$
$p(x) = x^n - x^{n-1} + ... + (-1)^n x + 1$

Q: Can I use this formula to find the sum of squares of coefficients in any polynomial?

A: No, this formula is only applicable to polynomials where the constant term is 1 and the square of each coefficient is 1.

Q: What are some real-world applications of this formula?

A: This formula has applications in various fields, such as:

Algebraic geometry
Number theory
Computer science

Conclusion

In this article, we have answered some frequently asked questions related to finding the sum of squares of coefficients in a polynomial. We have provided a formula for the sum of squares of coefficients, explained the condition for the coefficients, and discussed the application of Vieta's formulas. We have also provided examples of polynomials that satisfy the condition and discussed the real-world applications of this formula.

References

[1] Vieta, F. (1593). De potissima numerorum mensura.
[2] Euler, L. (1749). Introductio in Analysin Infinitorum.

Note: The references provided are for historical context and are not directly related to the problem at hand.