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Under Which Operations are Polynomials Closed?

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by

Geneva Taylor

on 12 November 2014

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Transcript of Under Which Operations are Polynomials Closed?

Mega Mansion Progect
Operation: Addition
When two polynomials are added, the variables and the exponents do not change, so it's not possible to have an exponent not in the set (0,1, 2, 3, etc...). There is no division, so division by a variable is not possible, and there is a finite number of terms because the equation began with a finite number of terms. Therefore, the answer fits the definition of a polynomial.
ex: x^3 + 5x^3 + x^6 = x^6 + 6x^3
POLYNOMIALS ARE CLOSED UNDER ADDITION
Operation: Subtraction
When a polynomial is subtracted from another polynomial, the variables and exponents do not change, so having an exponent not in the set (0, 1, 2, 3, etc....) is not possible. There is no division, so division by a variable is not possible, and there is a finite number of terms in the answer because the equation began with a finite number of terms. Therefore, the answer fits the definition of a polynomial.
ex: (x^3 + 5x^4) - (x^6 + 11x^4) = -x^6 - 6x^4 + x^3
POLYNOMIALS ARE CLOSED UNDER SUBTRACTION
Under Which Operations Are Polynomials Closed?
Definition of Closed Under an Operation:
When you perform an operation on elements of a set and the answer is also in the set.
Definition of a Polynomial:
An expression that can contain exponents, variables, and constants, but cannot include division by a variable, an exponent not in the set (0, 1, 2, 3, etc...) or an infinite number of terms
Operation: Division
It is possible to divide a polynomial by a variable, and to have an exponent not in the set (0, 1, 2, 3, etc...). Therefore, the answer will not always fit the definition of a polynomial.
ex: polynomials -> x^5/x^7 = x^-2 <- not a polynomial
POLYNOMIALS ARE NOT CLOSED UNDER DIVISION
Operation: Multiplication
When multiplying polynomials, the variables do not change and the exponents are added together. Since exponents of polynomials are always whole numbers, the exponents will always be in the set (0, 1, 2, 3, etc...). The operation is multiplication, so division by a variable is not possible. There is a finite number of terms being multiplied, so the answer will be a finite number of terms. Therefore, the answer fits with the definition of a polynomial.
ex: (3x^2 + x) (5x^5 + 2x) = 15x^7 + 5x^6 +6x^3 + 2x^2
POLYNOMIALS ARE CLOSED UNDER MULTIPLICATION
CONCLUSION
Polynomials are closed under addition
Polynomials are closed under subtraction
Polynomials are closed under Multiplication
Polynomials are not closed under division
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