Identifying Polynomials- Which of the Following Expressions Qualify-

Which of the following are polynomials? Check all that apply.

Polynomials are a fundamental concept in algebra, consisting of a sum of terms, each of which is a constant multiplied by a non-negative integer power of a variable. In this article, we will explore some examples and determine which of the given expressions qualify as polynomials.

Firstly, let’s define what constitutes a polynomial. A polynomial is an expression of the form:

an x^n + an-1 x^(n-1) + … + a1 x + a0

where n is a non-negative integer, and a0, a1, …, an are constants. The highest power of the variable x is called the degree of the polynomial.

Now, let’s examine the given expressions and determine if they are polynomials:

1. 2x^3 – 5x^2 + 3x – 1
This expression is a polynomial because it consists of terms with non-negative integer powers of x, and the coefficients are constants. The degree of this polynomial is 3.

2. 3x^2 + 2x – 7
This expression is also a polynomial as it meets the criteria mentioned earlier. The degree of this polynomial is 2.

3. 5x – 3
This expression is a polynomial because it has a single term with a non-negative integer power of x and a constant coefficient. The degree of this polynomial is 1.

4. 2x^4 + 5x^3 – 3x^2 + 4x – 6
This expression is a polynomial, as it consists of terms with non-negative integer powers of x and constant coefficients. The degree of this polynomial is 4.

5. √x + 3
This expression is not a polynomial because it contains a term with a fractional exponent (√x). Polynomials must have integer exponents.

6. 2x^3 + 5x^2 – 3x + 4
This expression is a polynomial because it meets the criteria mentioned earlier. The degree of this polynomial is 3.

7. 2x^2 + 5x – 3
This expression is a polynomial, as it has terms with non-negative integer powers of x and constant coefficients. The degree of this polynomial is 2.

In conclusion, the polynomials from the given expressions are:

1. 2x^3 – 5x^2 + 3x – 1
2. 3x^2 + 2x – 7
3. 5x – 3
4. 2x^4 + 5x^3 – 3x^2 + 4x – 6
5. 2x^3 + 5x^2 – 3x + 4
6. 2x^2 + 5x – 3

Remember that polynomials are an essential part of algebra, and understanding their properties is crucial for solving various mathematical problems.