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Are they polynomials? How to identify them?

To determine whether the given expressions are polynomials, we need to verify if they meet the criteria for polynomials. A polynomial is an expression of the form:

\[ \sum_{m=0}^{n} a_m x^m \]

where \( a_m \) are coefficients (which can be any real number), and \( m \) and \( n \) are natural numbers (non-negative integers). Each term in a polynomial has a variable raised to a non-negative integer power.

Let's analyze the given expressions:

1. \( t^{5/2} + t \)
2. \( \left( \sqrt{x - 1} \right)^4 \)
3. \( \sqrt{x - 1} \)

### Analysis:

1. \( t^{5/2} + t \):
   - The term \( t^{5/2} \) has an exponent of \( 5/2 \), which is not an integer.
   - Therefore, \( t^{5/2} + t \) is **not** a polynomial because the exponent \( 5/2 \) is not a natural number.

2. \( \left( \sqrt{x - 1} \right)^4 \):
   - We simplify \( \left( \sqrt{x - 1} \right)^4 \) to \( (x - 1)^2 \).
   - Expanding this, we get \( x^2 - 2x + 1 \), which is a polynomial with exponents 2, 1, and 0 (all natural numbers).
   - Therefore, \( \left( \sqrt{x - 1} \right)^4 \) is a polynomial.

3. \( \sqrt{x - 1} \):
   - This expression is equivalent to \( (x - 1)^{1/2} \).
   - The exponent \( 1/2 \) is not an integer.
   - Therefore, \( \sqrt{x - 1} \) is **not** a polynomial.

### Summary:

- \( t^{5/2} + t \) is **not** a polynomial.
- \( \left( \sqrt{x - 1} \right)^4 \) is a polynomial.
- \( \sqrt{x - 1} \) is **not** a polynomial.

### How to Identify Polynomials:

To identify if an expression is a polynomial, check the following:
1. Each term should have a variable raised to a non-negative integer power.
2. The coefficients can be any real number.
3. Operations involved should only include addition, subtraction, and multiplication of variables and constants (no division by variables, no roots of variables unless they simplify to integer powers).

Would you like more details on any specific part or have any questions?

Here are 5 questions you might consider asking next:
1. What are the different types of polynomials based on their degrees?
2. Can you provide examples of non-polynomial expressions?
3. How do you perform polynomial long division?
4. What is the role of the leading coefficient in a polynomial?
5. How do you find the roots of a polynomial equation?

**Tip:** When dealing with polynomials, always check the exponents of the variables to ensure they are non-negative integers.

Learn how to identify polynomials and understand the criteria they must meet. Includes examples such as t^(5/2) + t, (sqrt(x - 1))^4, and sqrt(x - 1). Suitable for advanced high school students.

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