Enter The Degree Of The Polynomial Below.$7x^7 + 10x^4 + 4x^3 - 5x^{11} - 10x^6 - 6x^7$A. 11 B. 6 C. 10 D. 7

When it comes to polynomials, understanding the degree is crucial for various mathematical operations and applications. In this article, we will delve into the concept of the degree of a polynomial and provide a step-by-step guide on how to determine it.

What is the Degree of a Polynomial?

The degree of a polynomial is the highest power or exponent of the variable (usually x) in any of its terms. It is a fundamental concept in algebra and is used to classify polynomials based on their complexity.

Types of Polynomials Based on Degree

Polynomials can be classified into different types based on their degree:

• Monomial: A polynomial with only one term is called a monomial. The degree of a monomial is the exponent of the variable.
• Binomial: A polynomial with two terms is called a binomial. The degree of a binomial is the highest exponent of the variable in any of its terms.
• Trinomial: A polynomial with three terms is called a trinomial. The degree of a trinomial is the highest exponent of the variable in any of its terms.
• Polynomial of degree n: A polynomial with n terms is called a polynomial of degree n. The degree of a polynomial of degree n is the highest exponent of the variable in any of its terms.

Determining the Degree of a Polynomial

To determine the degree of a polynomial, we need to identify the term with the highest power of the variable. Here's a step-by-step guide:

1. Identify the terms: Break down the polynomial into its individual terms.
2. Determine the power of the variable: For each term, identify the power or exponent of the variable (usually x).
3. Find the highest power: Compare the powers of the variable in each term and identify the highest power.
4. Determine the degree: The degree of the polynomial is the highest power of the variable.

Example: Determining the Degree of a Polynomial

Let's consider the polynomial: $7x^7 + 10x^4 + 4x^3 - 5x^{11} - 10x^6 - 6x^7$

To determine the degree of this polynomial, we need to follow the steps outlined above:

1. Identify the terms: The polynomial has six terms: $7x^7$, $10x^4$, $4x^3$, $-5x^{11}$, $-10x^6$, and $-6x^7$.
2. Determine the power of the variable: For each term, the power of the variable is:
   • $7x^7$: 7
   • $10x^4$: 4
   • $4x^3$: 3
   • $-5x^{11}$: 11
   • $-10x^6$: 6
   • $-6x^7$: 7
3. Find the highest power: The highest power of the variable is 11.
4. Determine the degree: The degree of the polynomial is 11.

Conclusion

In conclusion, determining the degree of a polynomial is a straightforward process that involves identifying the term with the highest power of the variable. By following the steps outlined above, we can easily determine the degree of a polynomial and classify it based on its complexity.

Answer

The degree of the polynomial $7x^7 + 10x^4 + 4x^3 - 5x^{11} - 10x^6 - 6x^7$ is 11.

Final Answer

In this article, we will address some of the most frequently asked questions about determining the degree of a polynomial.

Q: What is the degree of a polynomial with no variable?

A: The degree of a polynomial with no variable is 0. This is because the power of the variable is not defined, and therefore, the degree is considered to be 0.

Q: What is the degree of a polynomial with a variable raised to a negative power?

A: The degree of a polynomial with a variable raised to a negative power is not defined. This is because the power of the variable is not a positive integer, and therefore, the degree is not defined.

Q: How do I determine the degree of a polynomial with multiple variables?

A: To determine the degree of a polynomial with multiple variables, you need to identify the term with the highest power of any of the variables. For example, if you have a polynomial with two variables, x and y, and the term with the highest power is $x^3y^2$, then the degree of the polynomial is 5.

Q: Can a polynomial have a degree of 0?

A: Yes, a polynomial can have a degree of 0. This is because the power of the variable is not defined, and therefore, the degree is considered to be 0.

Q: What is the difference between the degree of a polynomial and the degree of a term?

A: The degree of a polynomial is the highest power of the variable in any of its terms, while the degree of a term is the power of the variable in that specific term.

Q: How do I determine the degree of a polynomial with a coefficient?

A: The degree of a polynomial with a coefficient is the same as the degree of the polynomial without the coefficient. The coefficient only affects the value of the polynomial, not its degree.

Q: Can a polynomial have a degree of a fraction?

A: No, a polynomial cannot have a degree of a fraction. The degree of a polynomial is always a positive integer.

Q: How do I determine the degree of a polynomial with a variable raised to a fractional power?

A: The degree of a polynomial with a variable raised to a fractional power is not defined. This is because the power of the variable is not a positive integer, and therefore, the degree is not defined.

Q: What is the degree of a polynomial with a variable raised to a power that is not an integer?

A: The degree of a polynomial with a variable raised to a power that is not an integer is not defined. This is because the power of the variable is not a positive integer, and therefore, the degree is not defined.

Conclusion

In conclusion, determining the degree of a polynomial is a straightforward process that involves identifying the term with the highest power of the variable. By following the steps outlined above, we can easily determine the degree of a polynomial and classify it based on its complexity.

Final Answer

The final answer is: $\boxed{11}$