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Chapter 5: Problem 4

What is the degree of the polynomial \(2 x^{3}+8 x^{2}+3 x^{4}+2 x+10\) ?

Short Answer

Expert verified
The degree of the polynomial is 4.

Step by step solution

01

Identifying Terms of the Polynomial

Identify each term in the polynomial: \( 2x^3, 8x^2, 3x^4, 2x, \) and \( 10 \). Each term consists of a coefficient and a variable raised to some power.
02

Determining the Degree of Each Term

For each term, determine the exponent of the variable. The exponents are: 3 for \(2x^3\), 2 for \(8x^2\), 4 for \(3x^4\), 1 for \(2x\), and 0 for the constant \(10\).
03

Finding the Highest Degree

Compare the degrees of each term: 3, 2, 4, 1, and 0. The highest exponent among these is 4.
04

Conclusion

The degree of the polynomial is determined by the term with the highest exponent, which is \(3x^4\), thus the degree of the polynomial is 4.

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Key Concepts

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

polynomial terms
When we talk about polynomial terms, each term is a combination of numbers and variables that are multiplied together. The structure of a polynomial can be identified by looking at its terms, each separated by a plus (+) or minus (-) sign.

Polynomials are expressions of the form:
  • Coefficients: These are the constant numbers in each term. For example, in the term \(2x^3\), the coefficient is 2.
  • Variables: These are letters that represent numbers, such as \(x\) in the polynomial \(2x^3 + 8x^2 + 3x^4 + 2x + 10\).
  • Exponents: This is the power to which the variable is raised. In \(2x^3\), the exponent is 3.
By identifying the individual parts of each term, you build a strong foundation for understanding the broader function of the polynomial itself.
exponents in algebra
Exponents in algebra indicate how many times a number is multiplied by itself. In the context of polynomials, they show us the power to which the variable in a term is raised. Understanding exponents is crucial for solving many algebraic problems!

Here are a few key points:
  • Positive exponents: Numbers like \(x^3\) mean \(x\) is multiplied by itself three times.
  • Zero exponents: Any number raised to the zero power, such as \(x^0\), equals 1. This is important in polynomial terms like the constant "10", where the variable part is considered \(x^0\).
  • Comparing exponents: When viewing a polynomial, compare the exponents to determine characteristics like the degree of the polynomial.
Understanding exponents is key to navigating algebraic expressions, as they determine both the complexity and classification of polynomials.
highest degree term
The concept of the highest degree term in a polynomial is essential for determining the degree of the entire polynomial. A polynomial's degree is the largest exponent in any of its terms.

To find the highest degree term, you simply need to compare the exponents of all terms. Here is a simple process:
  • Identify each term and its exponent. For example, in \(2x^3 + 8x^2 + 3x^4 + 2x + 10\), check the exponents: 3, 2, 4, 1, and 0.
  • Recognize that constants, like "10", contribute an exponent of 0.
  • The highest exponent determines the degree of the polynomial. Thus, among the provided terms, \(3x^4\) has the highest degree, 4.
The highest degree term provides a quick snapshot of the polynomial's complexity and helps categorize the polynomial for further algebraic operations.

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Most popular questions from this chapter

Use your graphing calculator to sketch the graph of the quadratic polynomial \(p(x)=2 x^{2}-5 x-4 .\)

Simplify: \(\left(-5 a^{2} b+4 a b-3 a b^{2}\right)+\left(2 a^{2} b+7 a b-a b^{2}\right)\)

Given \(f(x)=2 x^{2}+9 x-5\) and \(g(x)=-x^{2}-4 x+3\), simplify \(f(x)+g(x)\).

Multiply: \(4\left(-2 y^{2}+3 y+5\right)\)

Sketch the graph of \(f(x)=-\frac{2}{3} x-2\)
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