• # 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) $(a+b+c)\,(a+b-c)$.

(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 $={{a}^{2}}+2ab+{{b}^{2}}-{{c}^{2}}$