Problem 105 Is it possible for a binomial to... [FREE SOLUTION]

Chapter 5: Problem 105

Is it possible for a binomial to have degree \(4 ?\) If so, give an example.

Short Answer

Expert verified

Yes, an example is \(5x^4 - 2.\)

Step by step solution

Understand the Concept of Degree

The degree of a polynomial is the highest power of the variable in the polynomial. So, for a binomial, we are looking for the highest exponent in one of its terms.

Define a Binomial

A binomial is a polynomial with exactly two terms. These terms can be constants, variables with exponents, or a combination of both.

Check for Degree 4

We need to check if a binomial can have a highest exponent of 4. Consider the binomial: \[x^4 + 3.\]Here, the highest exponent in the terms is 4, making the degree of this binomial 4.

Provide an Example

An example of a binomial with degree 4 is \[5x^4 - 2.\]This binomial has two terms, and the highest exponent (degree) is 4.

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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 essential to understand its behavior and properties. The degree is defined as the highest power of the variable present in the polynomial.
For instance, in the polynomial ax^n + bx^{n-1} + ... + c, the highest exponent or power of the variable x that appears is n.
This makes the degree of the polynomial n.
It's important to note that the degree dictates the most significant term and thus influences the polynomial's graph.
For example, if you have a polynomial: x^3 + 4x^2 - x + 6, the degree is 3 since the highest exponent in the terms is 3.

Binomial Definition

A binomial is a specific type of polynomial.
It comprises exactly two terms, making it unique.
These terms can be constants, variables raised to a power, or a combination of both.
For example: 3x^2 + 5 and 7y - 4 are both binomials, as each expression contains only two distinct terms.
Binomials often surface in various algebraic operations, from factoring to expanding polynomial products.

Highest Exponent in Terms

The highest exponent in a polynomial's terms is crucial for determining its degree.
This exponent is located by identifying the term within the polynomial that has the largest power.
For example, consider the binomial x^5 + 2x^2- Here, the term x^5 has the highest exponent.
This means the degree of the binomial is 5.
When dealing with polynomials, always look for the term with the largest exponent to quickly determine the degree and understand the polynomial's major characteristics.

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91影视

Is it possible for a binomial to have degree \(4 ?\) If so, give an example.

Short Answer

Step by step solution

Understand the Concept of Degree

Define a Binomial

Check for Degree 4

Provide an Example

Key Concepts

Degree of a Polynomial

Binomial Definition

Highest Exponent in Terms

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