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Chapter 5: Problem 111

What is the degree of a polynomial? Provide an example with your explanation.

Short Answer

Expert verified
The degree of a polynomial is the highest power of the variable in that polynomial. For example, the degree of the polynomial \(3x^4 - 2x^3 + 7x - 6\) is 4.

Step by step solution

01

Understanding the Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in that polynomial. It essentially tells us the highest number of roots the polynomial can have.
02

Provide an Example of Polynomial Degree

Take for example the polynomial \(3x^4 - 2x^3 + 7x - 6\). In this polynomial, the term with the highest power of \(x\) 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.

Understanding Polynomial Concepts
Polynomials are fundamental expressions in algebra that consist of variables and coefficients. These expressions involve the sum of terms, each term being a product of a constant coefficient and a variable raised to a non-negative integer power.
The structure of polynomials can vary from simple to complex, but their general form is easy to grasp. A basic polynomial looks something like this:
  • Constant: a single number, like 5
  • Monomial: a term with a coefficient and a variable, such as \(3x\)
  • Binomial: a polynomial with two terms, for example, \(x^2 - 4x\)
  • Trinomial: a polynomial with three terms like \(x^2 - 4x + 4\)
Sometimes, polynomials are given as a combination of different terms with different powers and coefficients. Recognizing this form helps identify the degree of a polynomial, which is central to our understanding.
Working with Variable Powers
When dealing with polynomials, it's crucial to understand the concept of variable powers. These powers determine the degree of each term in a polynomial and help specify its behavior.
A variable power is simply the exponent attached to the variable in any term, which tells you how many times the variable is multiplied by itself. For example, in the term \(x^3\), the power of the variable is 3, signifying \(x \times x \times x\).
  • The power affects the shape of the polynomial's graph.
  • Higher powers indicate steeper or more complex curves.
In the polynomial \(3x^4 - 2x^3 + 7x - 6\), different terms offer different powers: 4, 3, 1, and 0 respectively, with the highest being 4. This highest power determines the degree of the polynomial.
Exploring the Roots of a Polynomial
Roots of a polynomial are the values of the variable that make the polynomial equal to zero. These are also sometimes referred to as solutions or zeros of the polynomial equation. Studying roots is essential because they offer insights into the behavior and solution of polynomial equations.
For any polynomial, the number of roots it can have is related to its degree. Specifically, a polynomial of degree \(n\) can have up to \(n\) roots. However, not all roots have to be real numbers; some can be complex.
  • Some polynomials have all real roots, like \(x^2 - 5 = 0\).
  • Others can have a mix of real and complex roots.
  • Finding roots often involves factoring, using the quadratic formula, or numerical methods for higher degree polynomials.
In analyzing the polynomial \(3x^4 - 2x^3 + 7x - 6\), we acknowledge it can have up to four roots, aligning with its degree.

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Most popular questions from this chapter

Solve: \(8-6 x>4 x-12\). (Section \(2.7,\) Example 7 )

perform the indicated operations. $$\begin{aligned} &\left[\left(7 y^{2}-4 y+2\right)-\left(12 y^{2}+3 y-5\right)\right]-\left[\left(5 y^{2}-2 y-8\right)+\left(-7 y^{2}+10 y-13\right)\right] \end{aligned}$$

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. Each statement applies to the division problem $$\frac{x^{3}+1}{x+1}$$ The purpose of writing \(x^{3}+1\) as \(x^{3}+0 x^{2}+0 x+1\) is to keep all like terms aligned.

$$\text { Graph: } y=\frac{1}{3} x+2$$

Subtract \(x^{3}-2 x^{2}+2\) from the sum of \(4 x^{3}+x^{2}\) and \(-x^{3}+7 x-3\)
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