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Chapter 3: Problem 10

Find the degree and leading coefficient of each polynomial \(5 x^{6}\)

Short Answer

Expert verified
The degree is 6 and the leading coefficient is 5.

Step by step solution

01

Identify the Degree

The degree of a polynomial is the highest power of the variable within the polynomial. In the polynomial \(5x^6\), the exponent on \(x\) is 6. Thus, the degree of this polynomial is 6.
02

Determine the Leading Coefficient

The leading coefficient is the coefficient of the term with the highest degree. In this polynomial, \(5x^6\), the coefficient of the term \(x^6\) is 5. Therefore, the leading coefficient is 5.

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Key Concepts

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

Degree of a Polynomial
Understanding the degree of a polynomial is quite straightforward once you recognize it as the highest exponent present in any single term of the polynomial. For instance, in the polynomial \(5x^6\), the highest power of \(x\) is 6, which makes it the degree of the polynomial. Identifying this degree is essential when analyzing the behavior of polynomials, as it tells us about the polynomial's curve and how it will grow as \(x\) increases. Additionally, knowing the degree is crucial when performing operations like addition, subtraction, or division with other polynomials.
Leading Coefficient
The leading coefficient is the coefficient in front of the term with the highest degree of the polynomial. In the polynomial \(5x^6\), the coefficient of the term \(x^6\) is 5, which makes this number the leading coefficient. Recognizing the leading coefficient is significant because:
  • It gives insight into the behavior of the polynomial's graph, such as whether the ends of the graph shoot up or down.
  • When graphing polynomials, it particularly impacts the steepness and direction of the curve.
Keep in mind that the leading coefficient is not just any number but specifically refers to the one accompanying the term with the highest power.
Exponents
Exponents are an integral part of polynomials, acting as indicators of how many times a number, known as the base, is multiplied by itself. In the term \(x^6\), the 6 is an exponent, showing that \(x\) is multiplied by itself six times. Working with exponents:
  • Helps simplify complex polynomial expressions, allowing a more manageable form.
  • Facilitates polynomial differentiation and integration processes.
Understanding exponents is foundational to mastering polynomials, and they have numerous applications from algebra to calculus. Whenever you see a term like \(x^n\), remember that the exponent \(n\) plays a critical role in defining the term's contribution to the polynomial's overall degree and character.

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Most popular questions from this chapter

Write an equation for a rational function with the given characteristics Vertical asymptotes at \(x=-4\) and \(x=-5\) \(x\) intercepts at (4,0) and (-6,0)\(\quad\) Horizontal asymptote at \(y=7\)

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Write an equation for a rational function with the given characteristics Vertical asymptotes at \(x=-3\) and \(x=6\) \(x\) intercepts at (-2,0) and (1,0) Horizontal asymptote at \(y=-2\)

Use your calculator or other graphing technology to solve graphically for the zeros of the function \(g(x)=x^{3}-6 x^{2}+x+28\)

Find the vertical and horizontal intercepts of each function \(f(t)=2(t-1)(t+2)(t-3)\)
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