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Answer:
Step-by-step explanation:
a) 1) Polynomial is closed under addition. In polynomial addition, only the coefficient change and variables and it powers does not change.
Example: Add 3x² - 2x + 7 and -2x² - 5x + 3
3x² - 2x + 7 - 2x² - 5x + 3 = 3x² -2x² - 2x - 5x + 7 + 3
= x² - 7x + 10
2)Polynomial is closed under addition. In polynomial subtraction, only the coefficient change and variables and it powers does not change.
Example: Subtract 3x - 2 from 4x + 2
4x + 2 - (3x - 2) = 4x + 2 - 3x + 2 = 7x + 4
3) Polynomial is closed under multiplication. In this, exponents of the variables are added according to the exponent rule. As the exponent of polynomials are whole numbers and whole numbers are closed under addition, the result will be a whole number.
-2x³ * 3x² =
4) Polynomial is not closed under multiplication. In division, we may get a negative exponent. So, it is not closed under division.
Addition subtraction multiplication Division
Natural numbers yes not closed Yes Not closed
Integers Yes Yes Yes not closed
Rational numbers Yes Yes Yes not closed
Irrational numbers not closed not closed not closed not closed