Answer:
Polynomial
[tex]x^{2}+3x^{5}-6x+12x^{3}[/tex]
[tex]x^{4}+3x^{3}-5x^{2}-7x+20[/tex]
[tex]x^{3}-2x^{2}+x-12[/tex]
Not a polynomial
[tex]\frac{5}{x^{3}}-\frac{1}{x^{2}}+\frac{8}{x}}-40[/tex]
[tex]x^{-3}-x^{-2}-6x^{-1}+8[/tex]
[tex]4\sqrt[3]{x}-\sqrt{x}-20[/tex]
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 [tex]\frac{5}{x^{3}}-\frac{1}{x^{2}}+\frac{8}{x}}-40[/tex] has division operation
∴ [tex]\frac{5}{x^{3}}-\frac{1}{x^{2}}+\frac{8}{x}}-40[/tex] ⇒ Not a polynomial
∵ The expression [tex]x^{2}+3x^{5}-6x+12x^{3}[/tex] has positive integer
exponents and the operations in it are addition, subtraction
∴ [tex]x^{2}+3x^{5}-6x+12x^{3}[/tex] ⇒ Polynomial
∵ The expression [tex]x^{4}+3x^{3}-5x^{2}-7x+20[/tex] has positive integer
exponents and the operations in it are addition, subtraction
∴ [tex]x^{4}+3x^{3}-5x^{2}-7x+20[/tex] ⇒ Polynomial
∵ The expression [tex]x^{-3}-x^{-2}-6x^{-1}+8[/tex] has negative integer
exponents
∴ [tex]x^{-3}-x^{-2}-6x^{-1}+8[/tex] ⇒ Not a polynomial
∵ The expression [tex]4\sqrt[3]{x}-\sqrt{x}-20[/tex] has non integer exponents
([tex]\sqrt[3]{x}=x^{\frac{1}{3}}[/tex] and [tex]\sqrt{x}=x^{\frac{1}{2}}[/tex])
∴ [tex]4\sqrt[3]{x}-\sqrt{x}-20[/tex] ⇒ Not a polynomial
∵ The expression [tex]x^{3}-2x^{2}+x-12[/tex] has positive integer
exponents and the operations in it are addition, subtraction
∴ [tex]x^{3}-2x^{2}+x-12[/tex] ⇒ Polynomial
