Which Of The Following Is A Polynomial?A. X 4 + X − 4 + 16 X^4 + X^{-4} + 16 X 4 + X − 4 + 16 B. 1 X + 2 \frac{1}{x} + 2 X 1 ​ + 2 C. X 6 − 2 X − 4 + 3 \frac{x^6 - 2}{x^{-4} + 3} X − 4 + 3 X 6 − 2 ​ D. X 2 − 1 X^2 - 1 X 2 − 1

In mathematics, a polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication, and non-negative integer exponents. Polynomials are a fundamental concept in algebra and are used to model various real-world phenomena. In this article, we will examine four given expressions and determine which one is a polynomial.

What is a Polynomial?

A polynomial is a mathematical expression that consists 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 a polynomial is the highest power of the variable, which in this case is $n$.

Analyzing the Options

Now, let's analyze each of the given options to determine which one is a polynomial.

Option A: $x^4 + x^{-4} + 16$

This expression consists of three terms: $x^4$, $x^{-4}$, and $16$. The first two terms are variables with exponents, and the third term is a constant. However, the presence of the term $x^{-4}$ indicates that this expression is not a polynomial. In a polynomial, all exponents must be non-negative integers. Therefore, option A is not a polynomial.

Option B: $\frac{1}{x} + 2$

This expression consists of two terms: $\frac{1}{x}$ and $2$. The first term is a fraction with a variable in the denominator, and the second term is a constant. However, the presence of the fraction indicates that this expression is not a polynomial. In a polynomial, all terms must be combined using only addition, subtraction, and multiplication, and non-negative integer exponents. Therefore, option B is not a polynomial.

Option C: $\frac{x^6 - 2}{x^{-4} + 3}$

This expression consists of two terms: $\frac{x^6 - 2}{x^{-4} + 3}$. The numerator is a polynomial with a degree of $6$, and the denominator is a polynomial with a degree of $-4$. However, the presence of the fraction indicates that this expression is not a polynomial. In a polynomial, all terms must be combined using only addition, subtraction, and multiplication, and non-negative integer exponents. Therefore, option C is not a polynomial.

Option D: $x^2 - 1$

This expression consists of two terms: $x^2$ and $-1$. Both terms are variables with non-negative integer exponents, and the expression is combined using only addition and subtraction. Therefore, option D is a polynomial.

Conclusion

In conclusion, the only polynomial among the given options is option D: $x^2 - 1$. This expression consists of two terms, both of which are variables with non-negative integer exponents, and the expression is combined using only addition and subtraction.

What is the Degree of a Polynomial?

The degree of a polynomial is the highest power of the variable. In the case of option D, the degree is $2$, since the highest power of the variable is $2$.

How to Determine if an Expression is a Polynomial

To determine if an expression is a polynomial, follow these steps:

1. Check if all terms are combined using only addition, subtraction, and multiplication.
2. Check if all exponents are non-negative integers.
3. If both conditions are met, then the expression is a polynomial.

Examples of Polynomials

Here are some examples of polynomials:

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

Conclusion

In this article, we will answer some frequently asked questions about polynomials.

Q: What is a polynomial?

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

Q: What are the characteristics of a polynomial?

A: The characteristics of a polynomial are:

• All terms are combined using only addition, subtraction, and multiplication.
• All exponents are non-negative integers.
• The expression can have a variable or variables.

Q: What is the degree of a polynomial?

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

Q: How do I determine if an expression is a polynomial?

A: To determine if an expression is a polynomial, follow these steps:

1. Check if all terms are combined using only addition, subtraction, and multiplication.
2. Check if all exponents are non-negative integers.
3. If both conditions are met, then the expression is a polynomial.

Q: What are some examples of polynomials?

A: Here are some examples of polynomials:

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

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

A: A polynomial is a mathematical expression that consists of variables and coefficients combined using only addition, subtraction, and multiplication, and non-negative integer exponents. A rational expression is a mathematical expression that consists of a fraction with a polynomial in the numerator and a polynomial in the denominator.

Q: Can a polynomial have a variable in the denominator?

A: No, a polynomial cannot have a variable in the denominator. If a polynomial has a variable in the denominator, it is not a polynomial, but rather a rational expression.

Q: Can a polynomial have a negative exponent?

A: No, a polynomial cannot have a negative exponent. If a polynomial has a negative exponent, it is not a polynomial, but rather a rational expression.

Q: Can a polynomial have a fraction as a coefficient?

A: No, a polynomial cannot have a fraction as a coefficient. If a polynomial has a fraction as a coefficient, it is not a polynomial, but rather a rational expression.

Q: Can a polynomial have a variable as a coefficient?

A: Yes, a polynomial can have a variable as a coefficient. For example, in the polynomial $2x + 3$, the coefficient of $x$ is $2$, which is a variable.

Q: Can a polynomial have a constant as a coefficient?

A: Yes, a polynomial can have a constant as a coefficient. For example, in the polynomial $x^2 + 3x - 4$, the coefficient of $x^2$ is $1$, which is a constant.

Q: Can a polynomial have a zero coefficient?

A: Yes, a polynomial can have a zero coefficient. For example, in the polynomial $x^2 + 0x - 4$, the coefficient of $x$ is $0$.

Q: Can a polynomial have a negative coefficient?

A: Yes, a polynomial can have a negative coefficient. For example, in the polynomial $-x^2 + 3x - 4$, the coefficient of $x^2$ is $-1$.

Q: Can a polynomial have a fractional coefficient?

A: No, a polynomial cannot have a fractional coefficient. If a polynomial has a fractional coefficient, it is not a polynomial, but rather a rational expression.

Q: Can a polynomial have a variable as a constant?

A: No, a polynomial cannot have a variable as a constant. If a polynomial has a variable as a constant, it is not a polynomial, but rather a rational expression.

Q: Can a polynomial have a constant as a variable?

A: No, a polynomial cannot have a constant as a variable. If a polynomial has a constant as a variable, it is not a polynomial, but rather a rational expression.

Conclusion

In conclusion, a polynomial is a mathematical expression that consists 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 of the variable. To determine if an expression is a polynomial, check if all terms are combined using only addition, subtraction, and multiplication, and if all exponents are non-negative integers.