Problem 32 - Algebra Form and Function

Question

Write the polynomials in exercises in standard form $$a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0}$$ What are the values of the coefficients \(a_{0}, a_{1}, \ldots, a_{n} ?\) Give the degree of the polynomial. $$ \sqrt{7} $$

Step-by-Step Solution

Answer

Answer: The coefficients are \(a_{0} = \sqrt{7}\) and all other coefficients, \(a_{1}, a_{2}, \ldots, a_{n}\) are equal to 0. The degree of the polynomial is 0.

Step 1: Rewrite the expression in standard polynomial form

The given expression is: $$ \sqrt{7} $$ This expression does not have any variables, so it can be written in standard polynomial form as a constant term only: $$ 0x^0 + \sqrt{7} $$

Step 2: Identify the coefficients

Now that we have rewritten the expression in the standard polynomial form, it's easy to identify the coefficients: \(a_{0} = \sqrt{7}\) There are no other terms with variables, so all other coefficients are zero. In this case, there is only one term, so the degree of the polynomial is zero, and the coefficients are: $$ a_{0} = \sqrt{7}, \quad a_{1} = 0, \quad a_{2} = 0, \ldots, a_{n} = 0 $$

Step 3: Identify the degree of the polynomial

As there is only one term in this expression and it is a constant term without any variables, the degree of this polynomial is 0. In conclusion, for the given expression $$\sqrt{7},$$ the coefficients are \(a_{0} = \sqrt{7}\) and all other coefficients, \(a_{1}, a_{2}, \ldots, a_{n}\) are equal to 0. The degree of the polynomial is 0.

Key Concepts

Coefficients of Polynomials

In a polynomial, coefficients are the numerical factors in front of each term, which often include variables raised to different powers. Let's break this down with an example:

• In the polynomial expression \(3x^2 + 5x - 7\), the coefficients are \(3\), \(5\), and \(-7\).
• The coefficient of \(x^2\) is \(3\), the coefficient of \(x\) is \(5\), and the constant term is \(-7\).

Each term in a polynomial is structured in the form \(a_n x^n\), where \(a_n\) represents the coefficient, and \(x^n\) is the variable term with \(n\) being the exponent.
This indicates how much influence that particular term has in the overall equation.
When a polynomial is written without any variables, as in the case of \(\sqrt{7}\), the expression is considered to have only one term, \(a_0\).
In this example, \(a_0 = \sqrt{7}\), and all other coefficients corresponding to higher variable powers are zero.

Degree of a Polynomial

The degree of a polynomial is defined as the highest power of the variable present in the expression. Understanding the degree is an essential aspect because it helps determine the polynomial's behavior, such as its growth rate or the number of roots it can have.
Consider the polynomial \(4x^3 + 2x^2 - x + 9\): the degree is \(3\) because the highest exponent of \(x\) is 3.
The degree gives us a sense of the complexity of the polynomial function.When you have a constant polynomial, like \(\sqrt{7}\), there are no variables involved. In such cases, the degree is \(0\). This signifies that the polynomial is a constant function, and its graph is a horizontal line.

Constant Polynomial

A constant polynomial is a polynomial that does not contain any variables. It's represented by a single, unchanging number. Constant polynomials can be thought of as the simplest form of polynomials since they are simply a real number.
For example, the number 5 on its own (written as \(5x^0\)) is a constant polynomial.
In the exercise provided, \(\sqrt{7}\) is such a polynomial.

• The degree of a constant polynomial is 0, as there are no variable terms.
• Examples include numbers like \(5\), \(-3\), and \(\sqrt{7}\), all of which remain unchanged regardless of the value of any variable.

The main characteristic of a constant polynomial is that its graph will always be a straight horizontal line on the coordinate plane, reflecting the same value across all points, thus reflecting its unchanging nature over any inputs.