Finding the Sum of Squares of Coefficients

Oct 13, 2025 by windailys.com 73 views

Introduction

This article discusses a problem from a previous calculus book, where we are given a polynomial with specific properties and asked to find the sum of squares of its coefficients. The problem states that we have a polynomial $a_n = 1$ and $a_i^2 = 1$ , and we need to find the value of $c_1^2 + c_2^2 + c_3^2 + ... + c_n^2$ . We will explore the solution to this problem using Vieta's formulas and discuss the polynomials that satisfy these conditions.

Understanding the Problem

The problem provides us with a polynomial $a_n = 1$ and $a_i^2 = 1$ . This means that the constant term of the polynomial is 1, and the square of each coefficient is also 1. We are asked to find the sum of squares of the coefficients of the polynomial, which can be represented as $c_1^2 + c_2^2 + c_3^2 + ... + c_n^2$ .

Using Vieta's Formulas

Vieta's formulas provide a relationship between the coefficients of a polynomial and its roots. For a polynomial of degree $n$ , the formulas state that the sum of the roots is equal to the negation of the coefficient of the $(n-1)$ th term, divided by the leading coefficient. In this case, we have a polynomial with $a_n = 1$ , so the leading coefficient is 1.

Let's consider a polynomial of degree $n$ with roots $r_1, r_2, ..., r_n$ . According to Vieta's formulas, the sum of the roots is equal to the negation of the coefficient of the $(n-1)$ th term, which is $c_{n-1}$ . Therefore, we have:

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

Since $a_i^2 = 1$ , we know that the square of each coefficient is 1. This means that the coefficients can be represented as $c_i = \pm 1$ for each $i$ .

Finding the Sum of Squares of Coefficients

Now, let's consider the sum of squares of the coefficients, which is $c_1^2 + c_2^2 + c_3^2 + ... + c_n^2$ . We can rewrite this expression as:

c_1^2 + c_2^2 + c_3^2 + ... + c_n^2 = (c_1 + c_2 + ... + c_n)^2 - 2 \sum_{i < j} c_i c_j

The first term on the right-hand side is the square of the sum of the coefficients, which is equal to the square of the sum of the roots. The second term is the sum of the products of pairs of coefficients, which can be represented as $\sum_{i < j} c_i c_j$ .

Polynomials that Satisfy the Conditions

We are given that $a_i^2 = 1$ , which means that the square of each coefficient is 1. This implies that the coefficients can be represented as $c_i = \pm 1$ for each $i$ . Therefore, the polynomials that satisfy the conditions are those with coefficients that are either 1 or -1.

Conclusion

In this article, we discussed a problem from a previous calculus book, where we were given a polynomial with specific properties and asked to find the sum of squares of its coefficients. We used Vieta's formulas to relate the coefficients of the polynomial to its roots and found that the sum of squares of the coefficients is equal to the square of the sum of the roots minus twice the sum of the products of pairs of coefficients. We also discussed the polynomials that satisfy the conditions, which are those with coefficients that are either 1 or -1.

Polynomials with Coefficients 1 or -1

The polynomials that satisfy the conditions are those with coefficients that are either 1 or -1. These polynomials can be represented as:

p(x) = c_0 + c_1 x + c_2 x^2 + ... + c_n x^n

where $c_i = \pm 1$ for each $i$ . Some examples of such polynomials include:

$p(x) = 1 + x + x^2 + ... + x^n$
$p(x) = 1 - x + x^2 - ... + (-1)^n x^n$
$p(x) = 1 + x^2 + x^4 + ... + x^{2n}$

These polynomials have coefficients that are either 1 or -1, and they satisfy the conditions given in the problem.

Properties of Polynomials with Coefficients 1 or -1

Polynomials with coefficients that are either 1 or -1 have some interesting properties. For example, they are always monic, meaning that the leading coefficient is 1. They are also always symmetric, meaning that the coefficients are the same when the polynomial is written in reverse order.

Applications of Polynomials with Coefficients 1 or -1

Polynomials with coefficients that are either 1 or -1 have many applications in mathematics and computer science. For example, they can be used to represent Boolean functions, which are functions that take a binary input and produce a binary output. They can also be used to represent permutations, which are functions that take a set of elements and produce a new set of elements.

Conclusion

Q: What is the problem asking for?

A: The problem is asking for the sum of squares of the coefficients of a polynomial, given that the polynomial has specific properties.

Q: What are the specific properties of the polynomial?

A: The polynomial has the property that $a_n = 1$ and $a_i^2 = 1$ , where $a_i$ is the coefficient of the $i$ th term.

Q: How can we use Vieta's formulas to solve this problem?

A: Vieta's formulas provide a relationship between the coefficients of a polynomial and its roots. We can use these formulas to relate the coefficients of the polynomial to its roots and find the sum of squares of the coefficients.

Q: What is the relationship between the coefficients and the roots of the polynomial?

A: According to Vieta's formulas, the sum of the roots is equal to the negation of the coefficient of the $(n-1)$ th term, divided by the leading coefficient. In this case, we have a polynomial with $a_n = 1$ , so the leading coefficient is 1.

Q: How can we find the sum of squares of the coefficients?

A: We can find the sum of squares of the coefficients by using the formula:

c_1^2 + c_2^2 + c_3^2 + ... + c_n^2 = (c_1 + c_2 + ... + c_n)^2 - 2 \sum_{i < j} c_i c_j

Q: What is the significance of the polynomials with coefficients 1 or -1?

A: The polynomials with coefficients 1 or -1 are the polynomials that satisfy the conditions given in the problem. These polynomials have many interesting properties and applications.

Q: What are some examples of polynomials with coefficients 1 or -1?

A: Some examples of polynomials with coefficients 1 or -1 include:

$p(x) = 1 + x + x^2 + ... + x^n$
$p(x) = 1 - x + x^2 - ... + (-1)^n x^n$
$p(x) = 1 + x^2 + x^4 + ... + x^{2n}$

Q: What are some applications of polynomials with coefficients 1 or -1?

A: Polynomials with coefficients 1 or -1 have many applications in mathematics and computer science, including:

Representing Boolean functions
Representing permutations
Solving systems of linear equations

Q: How can we use the properties of polynomials with coefficients 1 or -1 to solve problems?

A: We can use the properties of polynomials with coefficients 1 or -1 to solve problems by representing the problem as a polynomial with coefficients 1 or -1 and then using the properties of these polynomials to find the solution.

Q: What are some common mistakes to avoid when working with polynomials with coefficients 1 or -1?

A: Some common mistakes to avoid when working with polynomials with coefficients 1 or -1 include:

Assuming that the polynomial is always monic
Assuming that the polynomial is always symmetric
Not considering the properties of the polynomial when solving a problem

Q: How can we verify the solution to a problem involving polynomials with coefficients 1 or -1?

A: We can verify the solution to a problem involving polynomials with coefficients 1 or -1 by checking that the solution satisfies the properties of the polynomial and that it is consistent with the given conditions.