Problem 29 - 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. $$ 2 x^{3}+x-2 $$

Step-by-Step Solution

Answer

Answer: The degree of the polynomial is 3, and the coefficients are: a₃ = 2, a₂ = 0, a₁ = 1, and a₀ = -2.

Step 1: Identify the degree of the polynomial

First, let's determine the degree of the polynomial. The degree is simply the highest exponent in the polynomial. In this case it's 3. So the degree of the polynomial is 3. We now have: Degree: 3

Step 2: Identify the coefficients of the polynomial

Now, let's identify each coefficient \(a_0, a_1, \ldots, a_n\). The coefficients are the numbers accompanying the different powers of x in the polynomial. In our case, they are: \(a_3 = 2\) \(a_2 = 0\) (since there is no \(x^2\) term in the polynomial) \(a_1 = 1\) (since there is only an x term with no coefficient written, implying it is 1) \(a_0 = -2\) (the constant term) So we have: \(a_3 = 2\) \(a_2 = 0\) \(a_1 = 1\) \(a_0 = -2\) In conclusion, the given polynomial is already in standard form with degree 3 and the coefficients are as follows: \(a_3 = 2, a_2 = 0, a_1 = 1,\) and \(a_0 = -2\).

Key Concepts

Standard Form

The standard form of a polynomial is a way of writing polynomials in a consistent and structured manner. This involves arranging the terms of a polynomial in descending order of their degree. The general template for a polynomial in standard form is expressed as: \[a_{n} x^{n} + a_{n-1} x^{n-1} + \cdots + a_{1} x + a_{0}\]This means that:

• The term with the highest exponent is written first.
• Subsequent terms are arranged in decreasing order of their exponents.
• The constant term (without any variable) is written last.

For our polynomial \(2x^{3} + x - 2\), it is already in standard form:- The term with the largest exponent, \(x^3\), is written first.- The middle term, \(x\), follows, representing \(x^1\).- Lastly, \(-2\) is the constant term. Arranging polynomials in this manner not only ensures clarity but also helps in quickly identifying key properties like the degree and coefficients.

Coefficients

Coefficients in a polynomial are the numerical factors that multiply the variable terms. They play a crucial role in describing the polynomial and solving related equations. In the polynomial written in standard form, such as:\[a_{n} x^{n} + a_{n-1} x^{n-1} + \cdots + a_{1} x + a_{0}\]The coefficients are represented by \(a_n, a_{n-1}, \ldots, a_1, a_0\). Each coefficient corresponds to a term in the polynomial:

• \(a_n\) is the coefficient of the term with the highest degree \(x^n\).
• \(a_0\) is the coefficient of the constant term or the term with \(x^0\).

In our example, \(2x^3 + x - 2\), the coefficients are:- \(a_3 = 2\) for the term \(2x^3\).- \(a_2 = 0\) since there is no \(x^2\) term.- \(a_1 = 1\) for the term \(x\).- \(a_0 = -2\) for the constant \term.Knowing the coefficients helps in evaluating, graphing, and transforming polynomials.

Degree of Polynomial

The degree of a polynomial is a simple yet informative property. It denotes the highest power of the variable present in the polynomial, indicating its complexity. In any given polynomial, such as:\[a_{n} x^{n} + a_{n-1} x^{n-1} + \cdots + a_{1} x + a_{0}\]The degree is the value of the largest exponent \(n\) for the variable \(x\). For the polynomial \(2x^3 + x - 2\), the degree can be easily identified as:

• The highest power of \(x\) is \(3\), from the term \(2x^3\).

Therefore, the degree of this polynomial is 3. Understanding the degree is fundamental for:- Predicting the behavior of polynomials at the extremes (as \(x\) approaches infinity).- Determining the shapes and features of its graph.Recognizing the degree helps set expectations for solutions and the levels of complexity encountered in problems involving polynomials.