Question

which of the following expressions are polynomials and which are not? why?
$x^{3}x^{4}-(3^{2}z^{5}cdot 18xz)$
answer
the expression is

Explanation:

Step1: Simplify the first term

When multiplying exponents with the same base, add exponents: $x^3 \cdot x^4 = x^{3+4} = x^7$

Step2: Simplify the second term

First calculate the constant coefficient: $3^2 = 9$, then multiply constants: $9 \cdot 18 = 162$. Multiply variables: $z^5 \cdot z = z^{5+1} = z^6$, so the term becomes $162xz^6$

Step3: Rewrite the full expression

The simplified expression is $x^7 - 162xz^6$

Step4: Verify polynomial criteria

All variables have non-negative integer exponents, and only uses addition/subtraction/multiplication of variables and constants.

Answer:

a polynomial
Because after simplification, the expression $x^7 - 162xz^6$ consists of terms with non-negative integer exponents on all variables, which meets the definition of a polynomial.