Which Algebraic Expression Is A Polynomial With A Degree Of 3?A. $4x^3 - \frac{2}{x}$ B. $2y^3 + 5y^2 - 5y$ C. $3y^3 - \sqrt{4y}$ D. $-xy_{\sqrt{6}}$

Which Algebraic Expression is a Polynomial with a Degree of 3?

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 Correct Polynomial with a Degree of 3

To determine which of the given algebraic expressions is a polynomial with a degree of 3, we need to examine each option carefully.

Option A: $4x^3 - \frac{2}{x}$

At first glance, this expression appears to be a polynomial with a degree of 3 because the highest power of the variable $x$ is 3. However, we need to consider the presence of the fraction $\frac{2}{x}$. In a polynomial, all terms must be in the form of a product of a coefficient and a power of the variable. The fraction $\frac{2}{x}$ can be rewritten as $\frac{2}{1}x^{-1}$, which is not a polynomial because it contains a negative exponent.

Option B: $2y^3 + 5y^2 - 5y$

This expression consists of three terms: $2y^3$, $5y^2$, and $-5y$. The highest power of the variable $y$ is 3, which means this expression is a polynomial with a degree of 3.

Option C: $3y^3 - \sqrt{4y}$

This expression also appears to be a polynomial with a degree of 3 because the highest power of the variable $y$ is 3. However, we need to consider the presence of the square root $\sqrt{4y}$. In a polynomial, all terms must be in the form of a product of a coefficient and a power of the variable. The square root $\sqrt{4y}$ can be rewritten as $\sqrt{4}\sqrt{y}$, which is not a polynomial because it contains a square root.

Option D: $-xy_{\sqrt{6}}$

This expression is not a polynomial because it contains a variable $y$ with a subscript $\sqrt{6}$, which is not a valid mathematical expression.

Conclusion

Based on our analysis, the correct answer is Option B: $2y^3 + 5y^2 - 5y$. This expression is a polynomial with a degree of 3 because the highest power of the variable $y$ is 3, and all terms are in the form of a product of a coefficient and a power of the variable.

Additional Examples and Practice

To reinforce your understanding of polynomials and their degrees, try the following examples:

• Identify the degree of the polynomial $x^2 + 2x - 3$.
• Determine whether the expression $2x^3 - \frac{1}{x}$ is a polynomial.
• Find the degree of the polynomial $y^4 - 2y^2 + 1$.

By practicing these examples and understanding the properties of polynomials, you will become more confident in your ability to identify and work with polynomials in mathematics.

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 determine whether an expression is a polynomial, examine each term to ensure it is in the form of a product of a coefficient and a power of the variable.
• A polynomial with a degree of 3 has the highest power of the variable equal to 3.

Further Reading

For more information on polynomials and their degrees, consult a mathematics textbook or online resource. Some recommended resources include:

• Khan Academy: Polynomials
• Math Open Reference: Polynomials
• Wolfram MathWorld: Polynomial

By continuing to learn and practice, you will become proficient in working with polynomials and their degrees in mathematics.
Polynomial Degree Q&A

Frequently Asked Questions About Polynomial Degrees

In this article, we will address some common questions and concerns about polynomial degrees. Whether you are a student, teacher, or simply interested in mathematics, this Q&A section will provide you with a better understanding of polynomial degrees and how to work with them.

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. 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.

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

A: To determine the degree of a polynomial, examine each term and identify the highest power of the variable. For example, in the polynomial $2x^3 + 3x^2 - 4x + 1$, the highest power of the variable $x$ is 3, so the degree is 3.

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

A: A polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. A non-polynomial expression, on the other hand, may contain division, roots, or other operations that are not allowed in polynomials. For example, the expression $2x^3 - \frac{1}{x}$ is not a polynomial because it contains a fraction.

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, which means it is always 0 or greater.

Q: How do I identify a polynomial with a degree of 0?

A: A polynomial with a degree of 0 is a constant polynomial, which means it is a number without any variables. For example, the polynomial $5$ has a degree of 0 because it is a constant.

Q: Can a polynomial have a degree of 1?

A: Yes, a polynomial can have a degree of 1. For example, the polynomial $2x + 3$ has a degree of 1 because the highest power of the variable $x$ is 1.

Q: How do I identify a polynomial with a degree of 2?

A: A polynomial with a degree of 2 is a quadratic polynomial, which means it is a polynomial with a highest power of 2. For example, the polynomial $x^2 + 2x - 3$ has a degree of 2 because the highest power of the variable $x$ is 2.

Q: Can a polynomial have a degree greater than 3?

A: Yes, a polynomial can have a degree greater than 3. For example, the polynomial $x^4 + 2x^3 - 3x^2 + x + 1$ has a degree of 4 because the highest power of the variable $x$ is 4.

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

A: To determine the degree of a polynomial with multiple variables, identify the highest power of any of the variables. For example, in the polynomial $x^2y + 2xy^2 + 3x^2y^2$, the highest power of any of the variables is 2, so the degree is 2.

Q: Can a polynomial have a degree of 0 with multiple variables?

A: No, a polynomial with multiple variables cannot have a degree of 0. The degree of a polynomial with multiple variables is always greater than 0.

Conclusion

In this Q&A section, we have addressed some common questions and concerns about polynomial degrees. Whether you are a student, teacher, or simply interested in mathematics, this article has provided you with a better understanding of polynomial degrees and how to work with them.

Additional Resources

For more information on polynomial degrees, consult a mathematics textbook or online resource. Some recommended resources include:

• Khan Academy: Polynomials
• Math Open Reference: Polynomials
• Wolfram MathWorld: Polynomial

By continuing to learn and practice, you will become proficient in working with polynomial degrees in mathematics.