Which algebraic expressions are polynomials? Check all that apply. 2x^3 - x^3y - 3x^2 + 6xy^2 + 5y - 2 - -x - x^3 - x^2 +
The Correct Answer and Explanation is:
Definition of a Polynomial
A polynomial is an algebraic expression that consists of variables and coefficients, combined using only addition, subtraction, and multiplication, and non-negative integer exponents.
That means:
- No variable in the denominator
- No negative or fractional exponents
- No square roots or other radicals with variables
- Each term must be a product of a constant and variables raised to whole number exponents
🧪 Let’s evaluate the given expressions:
1. 2×3−x3y−3×2+6xy2+5y−22x^3 – x^3y – 3x^2 + 6xy^2 + 5y – 22×3−x3y−3×2+6xy2+5y−2
- All terms involve variables raised to non-negative integer exponents
- All operations are addition/subtraction/multiplication
- ✅ This is a polynomial
2. −x−x3−x2-x – x^3 – x^2−x−x3−x2
- Variables are raised to powers 1, 3, and 2
- No fractions, radicals, or negative exponents
- ✅ This is a polynomial
(If there were more options listed, we would evaluate those as well.)
✅ Correct Answers:
- 2×3−x3y−3×2+6xy2+5y−22x^3 – x^3y – 3x^2 + 6xy^2 + 5y – 22×3−x3y−3×2+6xy2+5y−2
- −x−x3−x2-x – x^3 – x^2−x−x3−x2
🧠 Explanation
Polynomials are one of the foundational concepts in algebra. They are built from variables, constants (numbers), and exponents that are non-negative whole numbers. The structure of a polynomial is simple: it is a sum of terms, each of which is the product of a constant and one or more variables raised to whole-number powers.
For example, in the expression 2×3−x3y−3×2+6xy2+5y−22x^3 – x^3y – 3x^2 + 6xy^2 + 5y – 22×3−x3y−3×2+6xy2+5y−2, we observe that each term:
- Has variables like xxx and yyy
- The powers on those variables are 0 or positive integers (like 3, 2, 1)
- There are no square roots, division by a variable, or variables with negative or fractional exponents
So, this expression qualifies as a multivariable polynomial.
The second expression, −x−x3−x2-x – x^3 – x^2−x−x3−x2, is also a polynomial. Though it has negative coefficients, this is allowed. What matters is that the exponents on the variables are all non-negative integers. This expression is a single-variable polynomial in xxx.
If an expression included, for example, terms like x−1x^{-1}x−1, 1/x1/x1/x, x\sqrt{x}x, or x1/2x^{1/2}x1/2, it would not be a polynomial, because those involve negative or fractional exponents or roots.
In conclusion, polynomials are defined by the form of their terms, and both expressions listed meet that definition. Thus, they are polynomials.
