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Chapter 12: Problem 27

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

Short Answer

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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 by step solution

01

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: $$ \frac{1}{3}s^{13} $$
02

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: $$ \frac{1}{3} $$

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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 鈥� 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.

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Most popular questions from this chapter

Without expanding, what is the leading term of $$ (2 s+5)(3 s+1)(s-10) ? $$

The polynomial \(p(x)\) can be written in two forms: I. \(\quad p(x)=2 x^{3}-3 x^{2}-11 x+6\) II. \(p(x)=(x-3)(x+2)(2 x-1)\) Which form most readily shows (a) The zeros of \(p(x) ?\) What are they? (b) The vertical intercept? What is it? (c) The sign of \(p(x)\) as \(x\) gets large, either positive or negative? What are the signs? (d) The number of times \(p(x)\) changes sign as \(x\) increases from large negative to large positive \(x ?\) How many times is this?

Find two different polynomials of degree 3 with zeros \(1,2,\) and \(3 .\)

Evaluate the expressions in Problems \(51-54\) given that \(f(x)=2 x^{3}+3 x-3, \quad g(x)=3 x^{2}-2 x-4\) \(h(x)=f(x) g(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{0}\) $$ n $$

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. $$ (x-3)(2 x-1)(x-2) $$
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