question_answer 17)

                Simplify:                 (i) \[({{x}^{2}}-5)\,(x+5)\,+25\]                 (ii) \[({{a}^{2}}+5)\,({{b}^{3}}+3)+5\] (iii) \[(t+{{s}^{2}})\,({{t}^{2}}-s)\] (iv) \[(a+b)\,(c-d)\,+(a-b)\,(c+d)\]\[+2\,(ac+bd)\] (v) \[(x+y)\,(2x+y)\,+(x+2y)(x-y)\] (vi) \[(x+y)\,({{x}^{2}}-xy+{{y}^{2}})\] (vii) \[(1.5x-4y)\,(1.5x+4y+3)\,-4.5x+12y\] (viii) \[(a+b+c)\,(a+b-c)\].

Answer:

                (i) \[({{x}^{2}}-5)\,(x+5)\,+25\]  \[({{x}^{2}}-5)\,(x+5)\,+25\] \[={{x}^{2}}(x+5)5(x+5)+25\] \[={{x}^{3}}+5{{x}^{2}}-5x-25+25\] \[={{x}^{3}}+5{{x}^{2}}-5x\].                 (ii) \[({{a}^{2}}+5)\,({{b}^{3}}+3)+5\]      \[({{a}^{2}}+5)\,({{b}^{3}}+3)+5\] \[={{a}^{2}}({{b}^{3}}+3)\,+5({{b}^{3}}+3)+5\] \[={{a}^{2}}{{b}^{3}}+3{{a}^{2}}+5{{b}^{3}}+15+5\] \[={{a}^{2}}{{b}^{3}}+3{{a}^{2}}+5{{b}^{3}}+20\] (iii) \[(t+{{s}^{2}})\,({{t}^{2}}-s)\] \[(t+{{s}^{2}})\,({{t}^{2}}-s)\] \[=t({{t}^{2}}-s)\,+{{s}^{2}}({{t}^{2}}-s)\] \[={{t}^{3}}\,-ts\,+{{s}^{2}}\,{{t}^{2}}-{{s}^{3}}\] (iv) \[(a+b)\,(c-d)\,+(a-b)\,(c+d)\]\[+2\,(ac+bd)\] \[(a+b)\,(c-d)\,+(a-b)\,(c+d)\]\[+2\,(ac+bd)\] \[=a(c-d)\,+b(c-d)+a(c+d)\]\[-b(c+d)\,+2(ac+bd)\] \[=ac-ad\,+bc\,-bd+ac+ad-bc-bd\]\[+2ac\,+2bd\]                  \[=(ac+ac+2ac)+(ad-ad)+(bc-bc)\]\[+(2bd-bd-bd)\] \[=4ac\]                                                               |combining the like terms (v) \[(x+y)\,(2x+y)\,+(x+2y)(x-y)\] \[(x+y)\,(2x+y)\,+(x+2y)(x-y)\] \[=x(2x+y)+y(2x+y)+x(x-y)\]\[+2y(x-y)\] \[=2{{x}^{2}}+xy+2xy+{{y}^{2}}+{{x}^{2}}-xy+2xy\]\[-2{{y}^{2}}\] \[=(2{{x}^{2}}+{{x}^{2}})+(xy+2xy-xy+2xy)\]\[+({{y}^{2}}-2{{y}^{2}})\]   |combining the like terms                 \[=3{{x}^{2}}+4xy-{{y}^{2}}\] (vi) \[(x+y)\,({{x}^{2}}-xy+{{y}^{2}})\] \[(x+y)\,({{x}^{2}}-xy+{{y}^{2}})\] \[=x({{x}^{2}}-xy+{{y}^{2}})+y({{x}^{2}}-xy+{{y}^{2}})\] \[={{x}^{3}}-{{x}^{2}}y+x{{y}^{2}}+{{x}^{2}}y-x{{y}^{2}}+{{y}^{3}}\] \[={{x}^{3}}+({{x}^{2}}y-{{x}^{2}}y)+(x{{y}^{2}}-x{{y}^{2}})+{{y}^{3}}\]   |combining the like terms \[={{x}^{3}}+{{y}^{3}}\] (vii) \[(1.5x-4y)\,(1.5x+4y+3)\,-4.5x+12y\] \[(1.5x-4y)\,(1.5x+4y+3)\,-4.5x+12y\] \[=1.5x(1.5x\,+4y+3)-4y(1.5x+4y+3)\]\[-4.5x+12y\] \[=2.25{{x}^{2}}+6xy+4.5x-6xy-16{{y}^{2}}-12y\]\[-4.5x+12y\] \[=2.25{{x}^{2}}+(6xy-6xy)-16{{y}^{2}}\]\[+(4.5x-4.5x)\,+(12y-12y)\]      |combining the like terms \[=2.25{{x}^{2}}-16{{y}^{2}}\] (viii) \[(a+b+c)\,(a+b-c)\]. \[(a+b+c)\,(a+b-c)\] \[=a(a+b-c)\,+b(a+b-c)\,+c(a+b-c)\] \[={{a}^{2}}+ab-ac+ab+{{b}^{2}}-bc+ac\]\[+bc-{{c}^{2}}\] \[={{a}^{2}}+(ab+ab)+(ac-ac)+{{b}^{2}}\]\[+(bc-bc)\,-{{c}^{2}}\]                               |combining the like terms \[={{a}^{2}}+2ab+{{b}^{2}}-{{c}^{2}}\]