question_answer
1)
Add the following: (i) \[ab-bc,\,bc-ca,\,ca-ab\] (ii) \[a-b+ab,\,b-c+bc,\,c-a+ac\] (iii) \[2{{p}^{2}}{{q}^{2}}-3pq+4,\,5+7pq-3{{p}^{2}}{{q}^{2}}\] (iv) \[{{l}^{2}}+{{m}^{2}},\,{{m}^{2}}+{{n}^{2}},\,{{n}^{2}}+{{l}^{2}},\,2lm\,+2mn+2nl\].
Answer:
(i) \[ab-bc,\,bc-ca,\,ca-ab\] | \[ab-bc\] | | |
+ | | \[+bc-ca\] | |
+ | \[-ab\] | | \[ca\] |
0 |
(ii) \[a-b+ab,\,b-c+bc,\,c-a+ac\] | \[a-b+ab\] | | |
+ | \[+b\] | \[-c+bc\] | |
+ | \[-a\] | \[+c\] | \[+ac\] |
\[ab+bc\]\[+ac\] |
(iii) \[2{{p}^{2}}{{q}^{2}}-3pq+4,\,5+7pq-3{{p}^{2}}{{q}^{2}}\] | \[2{{p}^{2}}{{q}^{2}}-3pq+4\] |
+ | \[-3{{p}^{2}}{{q}^{2}}+7pq+5\] |
\[-{{p}^{2}}{{q}^{2}}+4pq+9\] |
(iv) \[{{l}^{2}}+{{m}^{2}},\,{{m}^{2}}+{{n}^{2}},\,{{n}^{2}}+{{l}^{2}},\,2lm\,+2mn+2nl\]. | \[{{l}^{2}}+{{m}^{2}}\] | | |
+ | | \[+{{m}^{2}}+{{n}^{2}}\] | |
+ | \[{{l}^{2}}\] | \[+{{n}^{2}}\] | |
+ | | | \[+2lm+2mn+2nl\] |
\[2{{l}^{2}}+2{{m}^{2}}+2{{n}^{2}}+2lm\,+2mn\,+2nl\] \[=2({{l}^{2}}+{{m}^{2}}+{{n}^{2}}+lm\,+mn\,+nl)\] |