Algebraic expressions are given. It is required to tell if they are polynomials, tell how many terms they have, and give reasons if they are not a polynomial.
Recall that polynomials are a monomial or a sum of monomials.
Recall also that monomials are a number, a variable, or a product of numbers and variables with whole-number exponents.
[tex](a)\text{ }x^2+4x-1[/tex]Notice that this satisfies the definition of a polynomial.
Hence, it is a polynomial and it has 3 terms.
[tex]\begin{gathered} (b)\text{ }12(x^3-6)=12x^3-72 \\ \end{gathered}[/tex]
Notice that this satisfies the definition of a polynomial.
Hence, it is a polynomial and it has 2 terms.
[tex](c)\text{ }\frac{2}{x}-3=2x^{-1}-3[/tex]
Notice that this expression does not satisfy the definition of a polynomial as it has a monomial with a negative number exponent variable, which is not a whole number.
Hence, it is not a polynomial, it has a term that has a variable with a negative exponent.
[tex](d)\text{ }9x^{40}[/tex]
This is a polynomial with just 1 term.
[tex](e)\text{ }6x^3-4[/tex]
This satisfies the definition of a polynomial.
Hence, it is a polynomial and it has 2 terms.
[tex](f)\text{ }3x^2-2x+1[/tex]
This also satisfies the definition of a polynomial as it only has monomials with variables of whole-number exponents.
Hence, it is a polynomial and it has 3 terms.
[tex](g)\text{ }4x^2-3x+2x^{-2}[/tex]
It is not a polynomial, it has a term that has a variable with a negative exponent.
[tex](h)\text{ }5+\frac{1}{2}x-\sqrt{3}x^2+x^3[/tex]
This also satisfies the definition of a polynomial as it only has monomials with variables of whole-number exponents.
Hence, it is a polynomial and it has 4 terms.
[tex](i)\text{ }3^{-1}x^2+5x-1[/tex]
This also satisfies the definition of a polynomial as it only has monomials with variables of whole-number exponents.
Hence, it is a polynomial and it has 3 terms.
