Question

describing the properties of polynomial addition
what do the properties of polynomial addition mean? complete each statement.
the closure property states that the sum of two polynomials is a
the commutative property states that changing the order of two or more terms
the value of the sum.
the associative property states that the way in which
the value.
constant
polynomial
variable

Explanation:

• Closure Property: By definition, when we add two polynomials, the result is always a polynomial. For example, if we have \(P(x)=2x + 3\) (a polynomial of degree 1) and \(Q(x)=x^{2}-x\) (a polynomial of degree 2), \(P(x)+Q(x)=x^{2}+x + 3\) is also a polynomial.
• Commutative Property: For any two polynomials \(A(x)\) and \(B(x)\), \(A(x)+B(x)=B(x)+A(x)\). Let \(A(x)=3x^{2}+2x\) and \(B(x)=x^{3}-x\). Then \(A(x)+B(x)=x^{3}+3x^{2}+x\) and \(B(x)+A(x)=x^{3}+3x^{2}+x\). Changing the order of the polynomials (terms) does not change the sum.
• Associative Property: For any three polynomials \(A(x)\), \(B(x)\), and \(C(x)\), \((A(x)+B(x))+C(x)=A(x)+(B(x)+C(x))\). Let \(A(x)=x\), \(B(x)=x^{2}\), \(C(x)=x^{3}\). \((A(x)+B(x))+C(x)=(x + x^{2})+x^{3}=x^{3}+x^{2}+x\) and \(A(x)+(B(x)+C(x))=x+(x^{2}+x^{3})=x^{3}+x^{2}+x\). Grouping the polynomials (terms) differently does not change the sum.

Answer:

• The closure property states that the sum of two polynomials is a polynomial.
• The commutative property states that changing the order of two or more terms does not change the value of the sum.
• The associative property states that the way in which terms are grouped does not change the value.