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Chapter 2: Problem 7

A polynomial function is written in _____ form when its terms are written in descending order of exponents from left to right.

Short Answer

Expert verified
A polynomial function is written in Standard Form when its terms are written in descending order of exponents from left to right.

Step by step solution

01

Understand Polynomial Function

A Polynomial function is a function that can be expressed in the form \(f(x) = a_nx^n + a_{n-1}x^{n-1} + . . . + a_2x^2 + a_1x + a_0\), where \(a_n, a_{n-1}, ... , a_2, a_1, a_0\) are constants and \(n\) is a nonnegative integer.
02

Identify the Property of Polynomial Function

The property of the polynomial function being described here is the arrangement of its terms. It is stated that the terms are written in descending order based on the degree of each term from left to right.
03

Determine the Form of Polynomial Function

A polynomial function arranged in this way, with terms written in descending order of degrees, is said to be in 'standard form' or 'descending order'.

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Key Concepts

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

Standard Form
The standard form of a polynomial is a specific way of writing the polynomial so that it's easy to read and understand. In this representation, each term of the polynomial is arranged in descending order, starting with the term that has the highest power of the variable, typically denoted as "x".

This form ensures that the term with the largest exponent comes first, followed by the next largest, and so on, until you reach the constant term with no variable.
  • The standard form of a polynomial is written as: \(a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0\)
  • The terms \(a_n, a_{n-1}, ... , a_0\) represent coefficients, which are constant numbers.
  • The list of terms \(n, n-1, ..., 1, 0\) are the exponents of the variable \(x\).

Thus, a polynomial given in standard form quickly shows both the highest power present and the number of terms. This enables easy comparisons between polynomials and a clear understanding of their structure.
Descending Order
Descending order in the context of a polynomial refers to sorting the terms from the highest to the lowest based on the exponents' values. This arrangement is crucial for understanding the impact of each term in the overall function.

When terms are ordered by decreasing powers of \(x\), like \(x^3\) before \(x^2\), it indicates the importance of each term in determining the polynomial's behavior.
  • Starting with the term with the highest degree ensures that the dominant component leads the expression.
  • Continuing in this manner with subsequent lower degrees reflects the influence of less dominant terms.

By always using descending order, the polynomial is ready for operations such as addition, subtraction, or finding derivatives, maintaining consistency within mathematical processes. This order is integral to identifying the polynomial's highest degree term clearly.
Degree of Polynomial
The degree of a polynomial is one of its most essential features. It tells you the highest power of the variable that is present in the polynomial expression. Identifying the degree is important because it provides insight into the characteristics and behavior of the polynomial function.

To determine the degree, look for the term with the largest exponent; this exponent is the degree of the polynomial.
  • For example, in the polynomial \(5x^4 + 2x^3 - x + 9\), the degree is 4, due to the largest exponent being 4 for the term \(5x^4\).
  • If only a constant term exists, like \(7\), the degree is 0 since constants can be represented as \(7x^0\).

The degree of a polynomial offers vital information about its graph, such as the number of possible turning points it might have and the end behavior of its graph.

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Most popular questions from this chapter

The cost \(C\) (in dollars) of supplying recycling bins to \(p \%\) of the population of a rural township is given by $$C=\frac{25,000 p}{100-p}, \quad 0 \leq p<100$$ (a) Use a graphing utility to graph the cost function. (b) Find the costs of supplying bins to \(15 \%, 50 \%,\) and 90\(\%\) of the population. (c) According to the model, would it be possible to supply bins to 100\(\%\) of the population? Explain.

Rational and Irrational Zeros, match the cubic function with the numbers of rational and irrational zeros. (a) Rational zeros: \(0 ;\) irrational zeros: 1 (b) Rational zeros: \(3 ;\) irrational zeros: 0 (c) Rational zeros: \(1 ;\) irrational zeros: 2 (d) Rational zeros: \(1 ;\) irrational zeros: 0 $$f(x)=x^{3}-x$$

IQ Scores The IQ scores for a sample of a class of returning adult students at a small northeastern college roughly follow the normal distribution $$y=0.0266 e^{-(x-100)^{2} / 450}, \quad 70 \leq x \leq 115$$ where \(x\) is the IQ score. (a) Use a graphing utility to graph the function. (b) From the graph in part (a), estimate the average number of hours per week a student uses the tutoring center.

Proof Prove that the complex conjugate of the product of two complex numbers \(a_{1}+b_{1} i\) and \(a_{2}+b_{2} i\) is the product of their complex conjugates.

Cost The ordering and transportation cost \(C\) (in thousands of dollars) for machine parts is $$C=100\left(\frac{200}{x^{2}}+\frac{x}{x+30}\right), \quad x \geq 1$$ where \(x\) is the order size (in hundreds). In calculus, it can be shown that the cost is a minimum when $$3 x^{3}-40 x^{2}-2400 x-36,000=0$$ Use a calculator to approximate the optimal order size to the nearest hundred units.
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