Decide if the following are polynomials.

a.

3x3+x2−x+3


To determine if the expression \( 3x^3 + x^2 - x + 3 \) is a polynomial, we need to check the following criteria:

1. **The exponents of the variable \( x \)**: All exponents must be non-negative integers (e.g., \( 0, 1, 2, 3, \ldots \)).

2. **The coefficients**: The coefficients of the terms can be any real number (positive, negative, zero, or a fraction).

### Analysis of the Expression

The expression given is:
\[
3x^3 + x^2 - x + 3
\]
- The term \( 3x^3 \) has an exponent of 3, which is a non-negative integer.
- The term \( x^2 \) has an exponent of 2, which is a non-negative integer.
- The term \( -x \) can be rewritten as \( -1x^1 \), where the exponent is 1, a non-negative integer.
- The constant term \( 3 \) can be considered as \( 3x^0 \), where the exponent is 0, a non-negative integer.

### Conclusion
Since all the exponents are non-negative integers and the coefficients are real numbers, the expression \( 3x^3 + x^2 - x + 3 \) **is a polynomial**.

Would you like to explore more about polynomials or have any specific questions? Here are some related questions that might be of interest:

1. What are the degrees of different polynomials?
2. Can polynomials have fractional exponents?
3. How do we classify polynomials based on their degree?
4. What are the standard forms of polynomials?
5. How do coefficients affect the shape of a polynomial graph?

**Tip:** When determining if an expression is a polynomial, always check if the exponents are non-negative integers.

Understand whether 3x^3 + x^2 - x + 3 qualifies as a polynomial. Explore the criteria involving exponents and coefficients in this detailed analysis. Suitable for high school students studying algebra and polynomial functions.

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