Which Algebraic Expression Is A Polynomial With A Degree Of $4$?A. $5x^4$B. $x^5 - 6x^4 + 14x^3 + X^2$C. $9x^4 - X^3$D. $2x^4 - 6x^4$

Which Algebraic Expression is a Polynomial with a Degree of 4?

Understanding Polynomials and Their Degrees

In mathematics, a polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. The degree of a polynomial is the highest power or exponent of the variable in the polynomial. For example, in the polynomial $ax^3 + bx^2 + cx + d$, the degree is 3 because the highest power of the variable $x$ is 3.

Identifying the Degree of a Polynomial

To identify the degree of a polynomial, we need to look at the term with the highest power of the variable. If there are multiple terms with the same highest power, we can combine them by adding or subtracting their coefficients. For example, in the polynomial $x^3 + 2x^3$, the degree is 3 because the highest power of the variable $x$ is 3.

Analyzing the Options

Now, let's analyze the options given in the problem:

A. $5x^4$

This option has only one term, $5x^4$, which means the degree of the polynomial is 4. The coefficient 5 is a constant, and it does not affect the degree of the polynomial.

B. $x^5 - 6x^4 + 14x^3 + x^2$

This option has multiple terms, and we need to identify the term with the highest power of the variable. In this case, the term $x^5$ has the highest power, which means the degree of the polynomial is 5, not 4.

C. $9x^4 - x^3$

This option also has multiple terms, and we need to identify the term with the highest power of the variable. In this case, the term $9x^4$ has the highest power, which means the degree of the polynomial is 4.

D. $2x^4 - 6x^4$

This option has two terms, $2x^4$ and $-6x^4$, which can be combined by adding their coefficients. The resulting term is $-4x^4$, which means the degree of the polynomial is 4.

Conclusion

Based on our analysis, the correct answer is either option A or D, both of which have a degree of 4. However, option A is a more straightforward example of a polynomial with a degree of 4, while option D is a combination of two terms that can be simplified to a polynomial with a degree of 4.

Key Takeaways

• A polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication.
• The degree of a polynomial is the highest power or exponent of the variable in the polynomial.
• To identify the degree of a polynomial, we need to look at the term with the highest power of the variable.
• If there are multiple terms with the same highest power, we can combine them by adding or subtracting their coefficients.

Additional Examples

Here are a few more examples of polynomials with different degrees:

• $x^2 + 3x + 2$ has a degree of 2.
• $2x^3 - 5x^2 + 3x + 1$ has a degree of 3.
• $x^4 - 2x^3 + 3x^2 - 4x + 1$ has a degree of 4.

Practice Problems

Try to identify the degree of the following polynomials:

• $x^3 + 2x^2 - 3x + 1$
• $2x^4 - 5x^3 + 3x^2 + 1$
• $x^2 + 3x - 2$

Answer Key

• $x^3 + 2x^2 - 3x + 1$ has a degree of 3.
• $2x^4 - 5x^3 + 3x^2 + 1$ has a degree of 4.
• $x^2 + 3x - 2$ has a degree of 2.

Conclusion

In conclusion, the degree of a polynomial is an important concept in mathematics that helps us understand the properties and behavior of polynomials. By identifying the term with the highest power of the variable, we can determine the degree of a polynomial and use it to solve problems and make predictions.
Polynomial Degree Q&A

Frequently Asked Questions About Polynomial Degrees

In this article, we will answer some of the most frequently asked questions about polynomial degrees. Whether you are a student, a teacher, or just someone who wants to learn more about polynomials, this article is for you.

Q: What is a polynomial?

A: A polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication.

Q: What is the degree of a polynomial?

A: The degree of a polynomial is the highest power or exponent of the variable in the polynomial.

Q: How do I identify the degree of a polynomial?

A: To identify the degree of a polynomial, you need to look at the term with the highest power of the variable. If there are multiple terms with the same highest power, you can combine them by adding or subtracting their coefficients.

Q: What is the difference between a polynomial and a non-polynomial expression?

A: A polynomial expression is one that consists of variables and coefficients combined using only addition, subtraction, and multiplication. A non-polynomial expression is one that includes other operations, such as division or exponentiation.

Q: Can a polynomial have a negative degree?

A: No, a polynomial cannot have a negative degree. The degree of a polynomial is always a non-negative integer.

Q: Can a polynomial have a fractional degree?

A: No, a polynomial cannot have a fractional degree. The degree of a polynomial is always a non-negative integer.

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 look at the term with the highest power of any of the variables. If there are multiple terms with the same highest power, you can combine them by adding or subtracting their coefficients.

Q: Can a polynomial have a degree of 0?

A: Yes, a polynomial can have a degree of 0. This is known as a constant polynomial, and it has the form $c$, where $c$ is a constant.

Q: Can a polynomial have a degree of 1?

A: Yes, a polynomial can have a degree of 1. This is known as a linear polynomial, and it has the form $ax + b$, where $a$ and $b$ are constants.

Q: Can a polynomial have a degree of 2?

A: Yes, a polynomial can have a degree of 2. This is known as a quadratic polynomial, and it has the form $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.

Q: Can a polynomial have a degree of 3?

A: Yes, a polynomial can have a degree of 3. This is known as a cubic polynomial, and it has the form $ax^3 + bx^2 + cx + d$, where $a$, $b$, $c$, and $d$ are constants.

Q: Can a polynomial have a degree of 4?

A: Yes, a polynomial can have a degree of 4. This is known as a quartic polynomial, and it has the form $ax^4 + bx^3 + cx^2 + dx + e$, where $a$, $b$, $c$, $d$, and $e$ are constants.

Conclusion

In conclusion, the degree of a polynomial is an important concept in mathematics that helps us understand the properties and behavior of polynomials. By answering these frequently asked questions, we hope to have provided you with a better understanding of polynomial degrees and how to work with them.

Additional Resources

If you want to learn more about polynomials and their degrees, here are some additional resources that you may find helpful:

• Khan Academy: Polynomials
• Mathway: Polynomials
• Wolfram MathWorld: Polynomials

Practice Problems

Try to identify the degree of the following polynomials:

• $x^3 + 2x^2 - 3x + 1$
• $2x^4 - 5x^3 + 3x^2 + 1$
• $x^2 + 3x - 2$

Answer Key

• $x^3 + 2x^2 - 3x + 1$ has a degree of 3.
• $2x^4 - 5x^3 + 3x^2 + 1$ has a degree of 4.
• $x^2 + 3x - 2$ has a degree of 2.