Hello! Let's conquer this problem by first, understanding what polynomials are.
Polynomials contain numbers(constants), variables, and sometimes exponents, which are all added, subtracted, multiplied, or divided by each other.
Rules include:
-variables cannot be the denominator in a polynomial, inside a square root symbol, or apart of an absolute function ([tex]\frac{1}{x} +3, \sqrt{4x}[/tex] , |x| is not a polynomial)
-the constants can be square roots, negative, fractions... ([tex]\sqrt{4} x, \frac{2x}{54} , -45[/tex] are all parts of a polynomial)
-the exponents must be non negative and numerical
Knowing this information, let's go through every choice:
A) Variables are not in the denominator, constants look good, and exponents are non negative and numerical. Don't let all those variables confuse you! Mixed variables still count as polynomials so it is a polynomial.
B) If you look closely, the exponent on the variable z is an x. As we stated earlier, the exponents have to be a number, so it is not a polynomial.
C) The variable z is under a square root, which does not qualify as a polynomail; therefore, it is not a polynomial.
D) Same scenario as A. Doesn't break any rules, therefore it is a polynomial.
E) Same scenario as A and D. It's a polynomial.
F) The whole equation is under an absolute value bracket; as we know, absolute value brackets around variables 'void' a polynomial, which makes it not a polynomial.
Therefore, A, D, and E are the polynomials in this problem.
