Question

drinks (4 pts each)- is $\frac{pi}{5}$ a monomial? explain your reasoning.- write a trinomial in standard form that has a leading coefficient of 3, a constant of 8 and a degree of 5.1which of the following is not a polynomial? justify your answer.$x^3 + x$ $x + 4^{-1}$$5^x$ $-8x^3y^2$

Explanation:

Step1: Define monomial, check $\frac{\pi}{5}$

A monomial is a single term that is a constant, a variable, or a product of constants and non-negative integer powers of variables. $\pi$ is a constant, so $\frac{\pi}{5}$ is a constant term (a single term with no variables, which fits the definition of a monomial).

Step2: Construct the required trinomial

A trinomial has 3 terms, standard form orders terms from highest degree to lowest. The leading term (degree 5) has coefficient 3, we add a middle term (degree between 1 and 4, e.g., degree 2 with coefficient 2), and the constant term 8.
Expression: $3x^5 + 2x^2 + 8$

Step3: Identify non-polynomial, justify

A polynomial consists of terms where variables have non-negative integer exponents, and variables are not in exponents.

• $x^3 + x$: All exponents are non-negative integers, so it is a polynomial.
• $x + 4^{-1}$: $4^{-1}=\frac{1}{4}$, which is a constant, so this is a polynomial.
• $5^x$: The variable $x$ is in the exponent, which violates polynomial rules.
• $-8x^3y^2$: All exponents are non-negative integers, so it is a polynomial.

Answer:

1. $\frac{\pi}{5}$ is a monomial because it is a single constant term, and constants are monomials.
2. $3x^5 + 2x^2 + 8$ (any valid trinomial with the given criteria is acceptable; this is one example)
3. $5^x$ is not a polynomial. Polynomials require variables to have non-negative integer exponents, but here the variable $x$ is the exponent itself, which does not meet the definition of a polynomial.