Which Algebraic Expression Is A Polynomial With A Degree Of $5$?A. $3x^5 + 8x^4y^2 - 9x^3y^3 - 6y^5$B. $2xy^4 + 4x^2y^3 - 6x^3y^2 - 7x^4$C. $8y^6 + Y^5 - 5xy^3 + 7x^2y^2 - X^3y - 6x^4$D. $-6xy^5 + 5x^2y^3 -

Which Algebraic Expression is a Polynomial with a Degree of $5$?

Understanding Polynomials and Their Degrees

In mathematics, a polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication, and non-negative integer exponents. The degree of a polynomial is the highest power or exponent of the variable in the polynomial. For example, in the polynomial $3x^5 + 8x^4y^2 - 9x^3y^3 - 6y^5$, the degree is $5$ because the highest power of the variable $x$ is $5$. In this article, we will explore which algebraic expression is a polynomial with a degree of $5$.

What is a Polynomial?

A polynomial is a mathematical expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication, and non-negative integer exponents. The general form of a polynomial is:

$$a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0$$

where $a_n, a_{n-1}, \ldots, a_1, a_0$ are coefficients, and $x$ is the variable. The degree of the polynomial is the highest power of the variable, which is $n$ in this case.

What is the Degree of a Polynomial?

The degree of a polynomial is the highest power of the variable in the polynomial. For example, in the polynomial $3x^5 + 8x^4y^2 - 9x^3y^3 - 6y^5$, the degree is $5$ because the highest power of the variable $x$ is $5$. The degree of a polynomial can be found by identifying the term with the highest power of the variable and taking the exponent of that term.

Evaluating the Options

Now that we have a good understanding of what a polynomial is and how to find its degree, let's evaluate the options given in the problem.

Option A: $3x^5 + 8x^4y^2 - 9x^3y^3 - 6y^5$

This polynomial has a degree of $5$ because the highest power of the variable $x$ is $5$. The term $3x^5$ has a degree of $5$, and there are no other terms with a higher degree.

Option B: $2xy^4 + 4x^2y^3 - 6x^3y^2 - 7x^4$

This polynomial has a degree of $4$ because the highest power of the variable $x$ is $4$. The term $-7x^4$ has a degree of $4$, and there are no other terms with a higher degree.

Option C: $8y^6 + y^5 - 5xy^3 + 7x^2y^2 - x^3y - 6x^4$

This polynomial has a degree of $6$ because the highest power of the variable $y$ is $6$. The term $8y^6$ has a degree of $6$, and there are no other terms with a higher degree.

Option D: $-6xy^5 + 5x^2y^3 - 3x^3y^2 + 2x^4y - x^5$

This polynomial has a degree of $5$ because the highest power of the variable $x$ is $5$. The term $-x^5$ has a degree of $5$, and there are no other terms with a higher degree.

Conclusion

Based on our analysis, the algebraic expression that is a polynomial with a degree of $5$ is:

• Option A: $3x^5 + 8x^4y^2 - 9x^3y^3 - 6y^5$
• Option D: $-6xy^5 + 5x^2y^3 - 3x^3y^2 + 2x^4y - x^5$

Both of these polynomials have a degree of $5$ because the highest power of the variable is $5$.
Polynomial Degree Q&A

Understanding Polynomials and Their Degrees

In mathematics, a polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication, and non-negative integer exponents. The degree of a polynomial is the highest power or exponent of the variable in the polynomial. In this article, we will answer some frequently asked questions about polynomial degrees.

Q: What is the degree of a polynomial?

A: The degree of a polynomial is the highest power of the variable in the polynomial. For example, in the polynomial $3x^5 + 8x^4y^2 - 9x^3y^3 - 6y^5$, the degree is $5$ because the highest power of the variable $x$ is $5$.

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

A: To find the degree of a polynomial, identify the term with the highest power of the variable and take the exponent of that term. For example, in the polynomial $2xy^4 + 4x^2y^3 - 6x^3y^2 - 7x^4$, the term with the highest power of the variable is $-7x^4$, so the degree of the polynomial is $4$.

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, and non-negative integer exponents. A non-polynomial expression is an expression that does not meet these criteria. For example, the expression $x^2 + 3x^{-1}$ is not a polynomial because it contains a negative exponent.

Q: Can a polynomial have a degree of zero?

A: Yes, a polynomial can have a degree of zero. A polynomial with a degree of zero is a constant polynomial, which is an expression that does not contain any variables. For example, the polynomial $5$ has a degree of zero because it does not contain any variables.

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 greater than or equal to zero.

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 term with the highest power of any of the variables and take the exponent of that term. For example, in the polynomial $3x^2y^3 + 4x^3y^2 - 6x^4y$, the term with the highest power of any of the variables is $-6x^4y$, so the degree of the polynomial is $4$.

Q: Can a polynomial have a degree of one?

A: Yes, a polynomial can have a degree of one. A polynomial with a degree of one is a linear polynomial, which is an expression of the form $ax + b$, where $a$ and $b$ are constants.

Q: Can a polynomial have a degree of two?

A: Yes, a polynomial can have a degree of two. A polynomial with a degree of two is a quadratic polynomial, which is an expression of the form $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.

Conclusion

In this article, we have answered some frequently asked questions about polynomial degrees. We have discussed what the degree of a polynomial is, how to find the degree of a polynomial, and the differences between polynomials and non-polynomial expressions. We have also discussed the properties of polynomials with different degrees, including constant polynomials, linear polynomials, and quadratic polynomials.