Problem 27 - Algebra Form and Function

Question

List the nonzero coefficients of the polynomials. $$ \frac{s^{13}}{3} $$

Step-by-Step Solution

Answer

Question: List the nonzero coefficients of the polynomial given by the fraction $$\frac{s^{13}}{3}$$. Answer: The nonzero coefficient is $$\frac{1}{3}$$.

Step 1: Rewrite the fraction as a polynomial

Since the given fraction is already in the simplest form, we don't need to do any simplifications or operations. We simply rewrite the given expression as a polynomial of s. The given expression is: $$ \frac{s^{13}}{3} $$ Rewriting it as a polynomial, we get the polynomial: $$ \frac{1}{3}s^{13} $$

Step 2: List the nonzero coefficients

Now that we have our polynomial written in the correct form, we can identify the nonzero coefficients. The polynomial consists of a single term, so we only have one coefficient to consider. In this case, the coefficient is \(\frac{1}{3}\), which is nonzero. Therefore, the list of nonzero coefficients is: $$ \frac{1}{3} $$

Key Concepts

Monomial

A monomial is the simplest form of a polynomial. It consists of just a single term. This term is composed of numbers, variables, and exponents, without any addition or subtraction connecting it to other terms. In mathematical terms, a monomial can be expressed as

• \( a \, x^n \)

where \( a \) is a constant known as the coefficient, \( x \) is the variable, and \( n \) is a non-negative integer exponent.
Monomials can be as straightforward as \( 5 \) or more complex, like \( 4xy^2z^3 \).
The key feature of a monomial is its singularity – it's a standalone term, unlike polynomials, which are sums of multiple such terms.

Polynomial Expression

A polynomial expression is a sum (or difference) of multiple monomials. It typically involves several terms where each term follows the form of a monomial. For example, a polynomial can look like:

• \( 3x^2 + 2x + 1 \)

In this expression, the terms are \( 3x^2 \), \( 2x \), and \( 1 \). A polynomial can have terms with different degrees, dictated by the highest power of the variable present.
When writing polynomials, they are often arranged in decreasing order of their exponents. This helps in understanding the degree of the polynomial, which is the highest exponent among the terms.
Polynomials are foundational in algebra and provide a framework for more complex mathematical calculations and functions.

Coefficient Identification

Identifying coefficients within polynomials is similar to unraveling the mystery of a math problem. Coefficients are the numerical parts of the terms in a polynomial.
Consider the polynomial expression \( 4x^3 + 3x^2 + 5 \). Here, the task of coefficient identification involves pinpointing these numbers:

• The coefficient of \( x^3 \) is 4.
• The coefficient of \( x^2 \) is 3.
• The standalone number 5 is technically the coefficient of \( x^0 \).

Finding coefficients is crucial because they help determine the contribution and significance of each term within the polynomial.
Understanding coefficients also enables students to solve equations and understand the relationships between different terms in algebraic expressions.