Question

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 two or more terms are grouped in a sum the value.

Explanation:

• Closure property: By definition, when we add two polynomials, the result is always a polynomial. For example, if \(P(x)=a_nx^n+\cdots+a_1x + a_0\) and \(Q(x)=b_mx^m+\cdots+b_1x + b_0\) (\(n\geq m\)), then \(P(x)+Q(x)=(a_nx^n+\cdots+a_1x + a_0)+(b_mx^m+\cdots+b_1x + b_0)\) is also of the form \(c_kx^k+\cdots+c_1x + c_0\) (where \(k = n\) in this case), which is a polynomial.
• Commutative property: For any two polynomials \(P(x)\) and \(Q(x)\), \(P(x)+Q(x)=Q(x)+P(x)\). Let \(P(x)=2x + 3\) and \(Q(x)=x^2+1\). Then \(P(x)+Q(x)=(2x + 3)+(x^2+1)=x^2+2x + 4\) and \(Q(x)+P(x)=(x^2+1)+(2x + 3)=x^2+2x + 4\). So, changing the order of the polynomials (terms) does not change the sum.
• Associative property: For any three polynomials \(P(x)\), \(Q(x)\), and \(R(x)\), \((P(x)+Q(x))+R(x)=P(x)+(Q(x)+R(x))\). Let \(P(x)=x\), \(Q(x)=x^2\), \(R(x)=x^3\). \((P(x)+Q(x))+R(x)=(x + x^2)+x^3=x^3+x^2+x\) and \(P(x)+(Q(x)+R(x))=x+(x^2+x^3)=x^3+x^2+x\). So, the way of grouping the polynomials (terms) does not change the sum.

Answer:

• The closure property: polynomial
• The commutative property: does not change
• The associative property: does not change