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Chapter 4: Problem 29

\(h(x)=x^4-2 x^3+3 x\)

Short Answer

Expert verified
The given polynomial is a fourth-degree polynomial with three terms: \(x^4\), \(-2x^3\), and \(3x\). The highest power is 4 and hence the degree is 4. The leading coefficient is 1. Without specific conditions provided, the polynomial function remains the same.

Step by step solution

01

Understand the given polynomial

The given polynomial \(h(x)=x^4-2 x^3+3 x\) is a fourth-degree polynomial. We need to examine the polynomial and the powers of \(x\). The coefficients of the powers of \(x\) are 1, -2, and 3 respectively. Find out the term with the highest degree and recognize that it would be \(x^4\).
02

Identify each term

From the given polynomial equation, we can see it is formed by three terms: \(x^4\), \(-2x^3\), and \(3x\). This analysis is important as it allows us to understand how the polynomial is constructed and how each term contributes to its structure.
03

Understand the degree and leading coefficient

The degree of the polynomial function is 4 as the highest exponent in the function is 4. The coefficient of the term with the highest degree will be the leading coefficient which in this instance is 1.
04

Apply the Polynomial Function

Applying the given coordinates or conditions, if any, to the polynomial formula will give the value of the function. However, without specific coordinates or conditions given, we would simply restate the equation as is.

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Key Concepts

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

Degree of a Polynomial
The degree of a polynomial is an important concept that determines its general behavior and graph shape. Simply put, the degree of a polynomial is the highest power of the variable in any term of the polynomial. For our given polynomial, \(h(x) = x^4 - 2x^3 + 3x\), the degrees of each term are as follows:
  • \(x^4\) has a degree of 4
  • \(-2x^3\) has a degree of 3
  • \(3x\) has a degree of 1
We determine that the degree of the polynomial is 4 because the largest exponent present is 4 from \(x^4\). This means the polynomial is a fourth-degree polynomial. Additionally, the degree affects features like the number of roots and the potential curves in its graph.
Leading Coefficient
The leading coefficient of a polynomial is the coefficient of the term with the highest degree. In our polynomial \(h(x) = x^4 - 2x^3 + 3x\), we know that:
  • The term with the highest degree is \(x^4\)
  • The coefficient next to \(x^4\) is 1
Therefore, the leading coefficient of the polynomial is 1.
The leading coefficient is significant because it can influence the end behavior of the polynomial's graph. For instance, since the leading coefficient is positive, the graph of \(h(x)\) will tend to go upwards as \(x\) moves towards positive or negative infinity.
Polynomial Terms
Polynomials are made up of terms, each consisting of variables raised to an exponent and multiplied by a coefficient. It's crucial to grasp the structure of a polynomial by identifying each term. Looking at \(h(x) = x^4 - 2x^3 + 3x\), we see:
  • \(x^4\): This is the highest-degree term, determining the polynomial's degree
  • \(-2x^3\): This term contributes to the shape and provides some curvature to the polynomial graph
  • \(3x\): Though it's of lower degree, each term is essential in constructing the final polynomial
Understanding each term helps in analyzing and graphing the polynomial.
Each term's exponent and coefficient play a vital role in how the polynomial behaves and is calculated or simplified.

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Most popular questions from this chapter

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In Exercises 43鈥�48, use Pascal鈥檚 Triangle to expand the binomial. \((2 t+4)^3\)

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\((4+i)(2-i)\)
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