To determine which of the given expressions are polynomials, we must first understand what constitutes a polynomial.
### Definition of a Polynomial
A polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.
### Examination of Each Expression
1. Expression: [tex]\( 9 \)[/tex]
This is a constant, which is considered a polynomial of degree 0.
Conclusion: This is a polynomial.
2. Expression: [tex]\( b + 6c \)[/tex]
This expression contains variables [tex]\( b \)[/tex] and [tex]\( c \)[/tex] with coefficients 1 and 6, respectively. Both variables have exponents of 1, which are non-negative integers.
Conclusion: This is a polynomial.
3. Expression: [tex]\( 36x^x - 2x^9 + 8x^7 \)[/tex]
Here, we see the term [tex]\( 36x^x \)[/tex]. The exponent [tex]\( x \)[/tex] is not a constant, making it not a non-negative integer. Polynomials require that exponents be non-negative integers.
Conclusion: This is not a polynomial.
4. Expression: [tex]\( -5x^4 + 7x^3 - 2x^2 + \frac{1}{2} \)[/tex]
This expression contains multiple terms of the form [tex]\( ax^n \)[/tex], where [tex]\( a \)[/tex] is a coefficient and [tex]\( n \)[/tex] is a non-negative integer. Specifically:
- [tex]\( -5x^4 \)[/tex] (where the exponent is 4)
- [tex]\( 7x^3 \)[/tex] (where the exponent is 3)
- [tex]\( -2x^2 \)[/tex] (where the exponent is 2)
- [tex]\( \frac{1}{2} \)[/tex] (a constant term, considered as [tex]\( \frac{1}{2}x^0 \)[/tex] with exponent 0)
Each term has a non-negative integer exponent.
Conclusion: This is a polynomial.
### Summary
- [tex]\( 9 \)[/tex] is a polynomial.
- [tex]\( b + 6c \)[/tex] is a polynomial.
- [tex]\( 36x^x - 2x^9 + 8x^7 \)[/tex] is not a polynomial.
- [tex]\( -5x^4 + 7x^3 - 2x^2 + \frac{1}{2} \)[/tex] is a polynomial.
Thus, the expressions that are polynomials are [tex]\( 9 \)[/tex], [tex]\( b + 6c \)[/tex], and [tex]\( -5x^4 + 7x^3 - 2x^2 + \frac{1}{2} \)[/tex].
