Question

Please help I don't understand this. I will give brainy and 10pts

Answer 1

Answer:

Polynomial

$x^{2}+3x^{5}-6x+12x^{3}$  

$x^{4}+3x^{3}-5x^{2}-7x+20$  

$x^{3}-2x^{2}+x-12$

Not a polynomial

$\frac{5}{x^{3}}-\frac{1}{x^{2}}+\frac{8}{x}}-40$

$x^{-3}-x^{-2}-6x^{-1}+8$

$4\sqrt[3]{x}-\sqrt{x}-20$

Step-by-step explanation:

A polynomial is an algebraic expression that has multiple terms made up of numbers and variables

A Polynomial has positive integer exponents and the operations in it are addition, subtraction, and multiplication  only

∵ The expression  $\frac{5}{x^{3}}-\frac{1}{x^{2}}+\frac{8}{x}}-40$ has division operation

∴  $\frac{5}{x^{3}}-\frac{1}{x^{2}}+\frac{8}{x}}-40$  ⇒ Not a polynomial

∵ The expression $x^{2}+3x^{5}-6x+12x^{3}$ has positive integer

   exponents and the operations in it are addition, subtraction

∴  $x^{2}+3x^{5}-6x+12x^{3}$ ⇒ Polynomial

∵ The expression  $x^{4}+3x^{3}-5x^{2}-7x+20$ has positive integer

   exponents and the operations in it are addition, subtraction

∴  $x^{4}+3x^{3}-5x^{2}-7x+20$ ⇒ Polynomial

∵ The expression  $x^{-3}-x^{-2}-6x^{-1}+8$ has negative integer

   exponents

∴  $x^{-3}-x^{-2}-6x^{-1}+8$ ⇒ Not a polynomial

∵ The expression $4\sqrt[3]{x}-\sqrt{x}-20$ has non integer exponents

  ($\sqrt[3]{x}=x^{\frac{1}{3}}$ and $\sqrt{x}=x^{\frac{1}{2}}$)

∴  $4\sqrt[3]{x}-\sqrt{x}-20$ ⇒ Not a polynomial

∵ The expression  $x^{3}-2x^{2}+x-12$  has positive integer

   exponents and the operations in it are addition, subtraction

∴  $x^{3}-2x^{2}+x-12$ ⇒ Polynomial