Which Expressions Are Polynomials?Select Each Correct Answer.A. $x^2+5 X^{\frac{1}{5}}$ B. $-x^2+5 X$ C. $6 X^2+5 X$ D. $-7 X^2+\frac{5}{3 X}$

In mathematics, a polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. Polynomials are a fundamental concept in algebra and are used to model various real-world phenomena. In this article, we will explore which expressions are polynomials and which are not.

What is a Polynomial?

A polynomial is an expression of the form:

$$a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$$

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

Examples of Polynomials

Here are some examples of polynomials:

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

Which Expressions Are Polynomials?

Now, let's examine the given expressions and determine which ones are polynomials.

A. $x^2+5 x^{\frac{1}{5}}$

This expression is not a polynomial because it contains a variable raised to a fractional exponent, $x^{\frac{1}{5}}$. In a polynomial, all exponents must be integers.

B. $-x^2+5 x$

This expression is a polynomial because it consists of variables and coefficients combined using only addition, subtraction, and multiplication. The exponents are integers, and there are no fractional exponents.

C. $6 x^2+5 x$

This expression is a polynomial because it consists of variables and coefficients combined using only addition, subtraction, and multiplication. The exponents are integers, and there are no fractional exponents.

D. $-7 x^2+\frac{5}{3 x}$

This expression is not a polynomial because it contains a fraction with a variable in the denominator, $\frac{5}{3 x}$. In a polynomial, all coefficients must be constants or variables raised to integer exponents.

Conclusion

In conclusion, the expressions that are polynomials are B and C. These expressions consist of variables and coefficients combined using only addition, subtraction, and multiplication, and the exponents are integers. The expressions A and D are not polynomials because they contain fractional exponents or a fraction with a variable in the denominator.

Key Takeaways

• A polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication.
• The exponents in a polynomial must be integers.
• A polynomial cannot contain fractional exponents or a fraction with a variable in the denominator.

Practice Problems

1. Determine which of the following expressions are polynomials:
   • $x^3 + 2x^2 - 3x + 1$
   • $x^2 + \frac{1}{x}$
   • $2x^4 - 3x^2 + 1$
2. Simplify the following polynomial expressions:
   • $x^2 + 3x + 2 + 2x^2$
   • $x^3 - 2x^2 + 3x - 1 + x^3$

Glossary

• Polynomial: An expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication.
• Degree: The highest power of the variable in a polynomial.
• Coefficient: A constant that is multiplied by a variable in a polynomial.
• Exponent: The power to which a variable is raised in a polynomial.
  Polynomial Q&A ==================

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

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

A: A polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. A rational expression, on the other hand, is an expression that can be written as the ratio of two polynomials.

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

A: No, a polynomial cannot have a variable in the denominator. In a polynomial, all coefficients must be constants or variables raised to integer exponents.

Q: Can a polynomial have a fractional exponent?

A: No, a polynomial cannot have a fractional exponent. In a polynomial, all exponents must be integers.

Q: What is the degree of a polynomial?

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

Q: How do you simplify a polynomial expression?

A: To simplify a polynomial expression, you can combine like terms by adding or subtracting the coefficients of the same variables.

Q: Can you give an example of a polynomial expression?

A: Here is an example of a polynomial expression:

$$x^2 + 3x + 2$$

This expression is a polynomial because it consists of variables and coefficients combined using only addition, subtraction, and multiplication.

Q: Can you give an example of a rational expression?

A: Here is an example of a rational expression:

$$\frac{x^2 + 3x + 2}{x + 1}$$

This expression is a rational expression because it can be written as the ratio of two polynomials.

Q: How do you determine if an expression is a polynomial or a rational expression?

A: To determine if an expression is a polynomial or a rational expression, you can look for the following characteristics:

• If the expression consists of variables and coefficients combined using only addition, subtraction, and multiplication, it is a polynomial.
• If the expression can be written as the ratio of two polynomials, it is a rational expression.

Q: Can you give some examples of polynomial expressions?

A: Here are some examples of polynomial expressions:

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

Q: Can you give some examples of rational expressions?

A: Here are some examples of rational expressions:

• $\frac{x^2 + 3x + 2}{x + 1}$
• $\frac{2x^3 - 4x^2 + x + 1}{x^2 + 1}$
• $\frac{x^4 + 2x^2 + 1}{x^2 + 2x + 1}$

Conclusion

In conclusion, polynomials and rational expressions are two important concepts in algebra. Polynomials are expressions consisting of variables and coefficients combined using only addition, subtraction, and multiplication, while rational expressions are expressions that can be written as the ratio of two polynomials. By understanding the characteristics of polynomials and rational expressions, you can determine which type of expression you are working with and simplify it accordingly.

Key Takeaways

• A polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication.
• A rational expression is an expression that can be written as the ratio of two polynomials.
• To determine if an expression is a polynomial or a rational expression, look for the characteristics mentioned above.
• Polynomials and rational expressions are used to model various real-world phenomena and are an essential part of algebra.

Practice Problems

1. Determine which of the following expressions are polynomials or rational expressions:
   • $x^2 + 3x + 2$
   • $\frac{x^2 + 3x + 2}{x + 1}$
   • $2x^3 - 4x^2 + x + 1$
2. Simplify the following polynomial expressions:
   • $x^2 + 3x + 2 + 2x^2$
   • $x^3 - 2x^2 + 3x - 1 + x^3$
3. Simplify the following rational expressions:
   • $\frac{x^2 + 3x + 2}{x + 1}$
   • $\frac{2x^3 - 4x^2 + x + 1}{x^2 + 1}$
   • $\frac{x^4 + 2x^2 + 1}{x^2 + 2x + 1}$