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Chapter 5: Problem 10

\(5 x^{3}-3 x^{2}+10\)

Short Answer

Expert verified
The degree of the polynomial \(5x^3 - 3x^2 + 10\) is 3, the leading coefficient is 5, and the constant term is 10.

Step by step solution

01

Understand the Polynomial

A polynomial is an expression of more than two algebraic terms, especially the sum of several terms that contain different powers of the same variable. In this case, the polynomial is \(5x^3 - 3x^2 + 10\). It consists of three terms.
02

Identify the Degree

The degree of the polynomial is the highest power of the variable that appears in the polynomial. Here, the highest power is 3, so the degree of the polynomial is 3.
03

Identify the Leading Coefficient

The leading coefficient is the coefficient of the term with the highest degree in the polynomial. The term with the highest degree here is \(5x^3\), so the leading coefficient is 5.
04

Identify the Constant Term

The constant term of a polynomial is the term that doesn't contain any variables. Here, the constant term is 10.

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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 to understand. It tells us about the highest power of the variable in the polynomial expression. In simpler terms, it's like checking which term of the polynomial has the biggest exponent attached to its variable. This helps us understand the overall "size" or complexity of the polynomial. In the polynomial \(5x^3 - 3x^2 + 10\), each term is a piece of our expression:
  • \(5x^3\): Here, the exponent is 3.
  • \(-3x^2\): Here, the exponent is 2.
  • \(10\): This is a constant, with no variable at all, so its implied power is 0.
The degree in this case is 3, because that's the largest exponent among all the terms. Knowing the degree of a polynomial can help in predicting how it will behave in different scenarios, especially when you input values into it.
Leading Coefficient
The leading coefficient of a polynomial is another crucial element to identify. It acts like a beacon for the most significant term, which is the term having the highest degree. Within a given polynomial, it is always paired with this term.

In our example polynomial \(5x^3 - 3x^2 + 10\), let's pinpoint the leading term and its coefficient:
  • The term \(5x^3\) is the leading term since its variable has the highest exponent which is 3.
  • The number paired with this leading term, which is 5, is called the leading coefficient.
The leading coefficient can impact properties such as the width or direction of a graph of the polynomial equation when it is plotted. Understanding this, you can learn how expressions grow and shrink, and how they behave at extremes.
Constant Term
The constant term in a polynomial is all about simplicity鈥攊t鈥檚 the part of the expression that stands alone without any variables attached. This term represents a fixed value that doesn鈥檛 change, no matter what the input might be for the variable in the polynomial.

Let鈥檚 find the constant term in our polynomial example \(5x^3 - 3x^2 + 10\):
  • The term \(10\) is the constant term; it doesn鈥檛 have an \(x\) or any other variable beside it.
Constant terms play an important role when evaluating the polynomial at zero or establishing a baseline or intercept when graphing the polynomial. Understanding how to spot them helps simplify polynomial evaluation and analysis.

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