Problem 38 - 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. $$ 1+x+\frac{x^{2}}{2}+\frac{x^{3}}{6}+\frac{x^{4}}{24}+\frac{x^{5}}{120} $$

Step-by-Step Solution

Answer

The coefficients of the polynomial are \(a_{0} = 1,\) \(a_{1} = 1,\) \(a_{2} = \frac{1}{2},\) \(a_{3} = \frac{1}{6},\) \(a_{4} = \frac{1}{24},\) and \(a_{5} = \frac{1}{120}.\) The degree of the polynomial is 5.

Step 1: \(a_{5} = \frac{1}{120},\)

\(a_{4} = \frac{1}{24},\)

Step 2: \(a_{3} = \frac{1}{6},\)

\(a_{2} = \frac{1}{2},\)

Step 3: \(a_{1} = 1,\)

\(a_{0} = 1.\) #Step 2: Find the degree of the polynomial# Since the highest power of \(x\) present in the polynomial is 5, the degree of the polynomial is 5. #Final Answer# Coefficients of the polynomial are: \(a_{0} = 1,\) \(a_{1} = 1,\) \(a_{2} = \frac{1}{2},\) \(a_{3} = \frac{1}{6},\) \(a_{4} = \frac{1}{24},\) and \(a_{5} = \frac{1}{120}.\) The degree of the polynomial is 5.

Key Concepts

Standard Form

When working with polynomials, presenting them in **standard form** is crucial. The standard form is a way of organizing a polynomial by descending powers of the variable, which makes it easier to identify key features like coefficients and the degree. Here's the general template for the standard form of a polynomial:

• \( a_{n}x^{n} + a_{n-1}x^{n-1} + \cdots + a_{1}x + a_{0} \)

In this polynomial, the terms are ordered from the highest power of \(x\) down to the constant term.Using this format, the polynomial from the exercise:

• \( 1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \frac{x^4}{24} + \frac{x^5}{120} \)

is arranged correctly, as each term decreases in power from \(x^5\) to \(x^0\). Staying organized helps to spot coefficients and makes calculating the degree straightforward.

Coefficients

Coefficients are the numerical factors that multiply the terms of a polynomial. Each term in a polynomial comprises a coefficient and a variable part raised to an exponent. Understanding coefficients is essential as they indicate how much each power of the variable contributes to the polynomial.For the given polynomial, the coefficients are:

• \( a_{5} = \frac{1}{120} \) for the term \( \frac{1}{120}x^{5} \)
• \( a_{4} = \frac{1}{24} \) for the term \( \frac{1}{24}x^{4} \)
• \( a_{3} = \frac{1}{6} \) for the term \( \frac{1}{6}x^{3} \)
• \( a_{2} = \frac{1}{2} \) for the term \( \frac{1}{2}x^{2} \)
• \( a_{1} = 1 \) for the term \( 1x \)
• \( a_{0} = 1 \) for the constant term \( 1 \)

These coefficients tell us the contribution of each power of \(x\) to the total polynomial value. Seeing them in context helps us understand the structure and behavior of the polynomial.

Degree of a Polynomial

The degree of a polynomial is determined by the highest power of the variable present in the polynomial. It's a critical attribute because it gives us insight into the polynomial's complexity and behavior at large values of \(x\).In our exercise, the polynomial is:

• \( 1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \frac{x^4}{24} + \frac{x^5}{120} \)

The highest power of \(x\) present is \(x^5\), so the degree of this polynomial is 5. Recognizing the degree helps when graphing or simplifying polynomials and makes identifying the leading term effortless.