3*x* − 5*x*3 + 7 is this expression a polynomial

Let's analyze the expression: 

\[ 3x - 5x^3 + 7 \]

To determine if an expression is a polynomial, we need to check the following criteria:

1. **The exponents of the variable \( x \) must be non-negative integers.** (For example: \( x^0, x^1, x^2, x^3, \) etc.)
2. **The coefficients can be any real number.**

### Analysis:

- The term \( 3x \) has \( x \) raised to the power of 1 (a non-negative integer).
- The term \( -5x^3 \) has \( x \) raised to the power of 3 (a non-negative integer).
- The constant term \( 7 \) can be viewed as \( 7x^0 \), where \( x^0 = 1 \) (so it’s valid).

Since all the exponents are non-negative integers and the coefficients are real numbers, **the expression is indeed a polynomial**.

Would you like more details or have any questions?

Here are 5 related questions you might find interesting:
1. What is the degree of this polynomial?
2. How do you classify polynomials by degree and number of terms?
3. How would you find the roots of this polynomial?
4. What is the difference between a monomial, binomial, and trinomial?
5. Can a polynomial have fractional or negative exponents?

**Tip:** The degree of a polynomial is determined by the highest exponent of \( x \) in the expression.

Explore whether the expression 3x - 5x^3 + 7 qualifies as a polynomial. Learn about the criteria for polynomials, including non-negative integer exponents and real-number coefficients. Suitable for high school students and those interested in algebraic expressions.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation