Question

How can the Associative, Commutative, and Distributive Properties be applied when performing operations on complex numbers and polynomials?

Answer 1

By using Associative, Commutative, and Distributive Properties in performing operations on complex numbers and polynomials, you would be able to simplify correctly and simply. To combine complex numbers, you must distribute any coefficients in front of the parentheses. Then identify andcombine like terms by combining the real number parts and the imaginary number parts separately.

Answer 2

Answer with explanation:

Complex Numbers

For,any two or more complex numbers , $Z_{1},Z_{2},Z_{3}$,

1. Commutative Property

    $Z_{1}+Z_{2}=Z_{2}+Z_{1}$

2. Associative Property

$Z_{1}+(Z_{2}+Z_{3})=(Z_{1}+Z_{2})+Z_{3},$

3.Distributive Property

$Z_{1}\times [Z_{2}+Z_{3}]=Z_{1} \times Z_{2}+Z_{1}\times Z_{3}$

Hence Complex number follows ,Associative, Commutative, and Distributive Property.

Polynomials ⇒

P=a x +b, Q=c x²+ t x+d, R=m x³+ n x²+ r x+s

1. Commutative Property

P+Q=a x+b+c x²+ t x+d

      =c x²+ t x+d+a x+b

    =Q+P

2. Associative Property

(P+Q)+R=(a x+b+c x²+ t x+d)+m x³+ n x²+ r x+s

   =a x+b+(c x²+ t x+d+m x³+ n x²+ r x+s)

    = P + (Q+R)

3.Distributive Property

P×(Q+R)

  (ax+b)×[c x²+ t x+d+m x³+ n x²+ r x+s]

=(ax+b)×[c x²+ t x+d]+(ax+b)[m x³+ n x²+ r x+s]

=P ×Q+P×R

Hence Polynomial follows ,Associative, Commutative, and Distributive Property.