question_answer 1)

                Multiply the binomials: (i) (2x + 5) and (4x - 3) (ii) (y - 8) and (3y - 4) (iii) (2.51 - 0.5 m) and (2.51 + 0.5m) (iv) (a + 3b) and (x + 5) (v) (2pq + 3q2) and (3pq - 2q2) (vi) \[\left( \frac{3}{4}{{a}^{2}}+3{{b}^{2}} \right)\,\] and \[4\left( {{a}^{2}}-\frac{2}{3}{{b}^{2}} \right)\]

Answer:

                (i) \[(2x+5)\] and \[(4x-3)\] \[(2x+5)\,\times (4x-3)\] \[=(2x)\times (4x-3)+5\times (4x-3)\] \[=(2x)\times (4x)-(2x)\times (3)+(5)\]\[\times (4x)-(5)\times (3)\] \[=8{{x}^{2}}-6x+20x-15\] \[=8{{x}^{2}}+(20x-6x)-15\]                         |Combining like terms \[=8{{x}^{2}}+14x-15\] (ii) (y ? 8) and (3y ? 4) \[(y-8)\times (3y-4)\] \[=y\times (3y-4)-8\times (3y-4)\] \[=(y)\times (3y)-(y)\times (4)-(8)\]\[\times (3y)+8\times 4\] \[=3{{y}^{2}}-4y-24y+32\] \[=3{{y}^{2}}-28y+32\]                  |Combining like terms (iii) \[(2.5\,l-0.5m)\] and \[(2.5\,l+0.5\,m)\] \[(2.5\,l+0.5\,m)\times (2.5\,l+0.5\,m)\] \[=(2.5\,l)\times (2.5\,l+0.5\,m)\]\[-(0.5\,m)\times (2.5l\,+0.5m)\] \[=(2.5\,l)\times (2.5\,l)+(2.5\,l)\]\[\times (0.5\,m)-(0.5\,m)\times (2.5\,l)\]\[-(0.5\,m)\,\times (0.5\,m)\] \[=6.25\,{{l}^{2}}+1.25\,lm-1.25\,ml\]\[-0.25\,{{m}^{2}}\]                                             |Combining like terms \[=6.25\,{{l}^{2}}-0.25\,{{m}^{2}}\] (iv) \[(a+3b)\] and \[(x+5)\] \[(a+3b)\times (x+5)\] \[=a\times (x+5)+(3b)\,\times (x+5)\] \[=(a)\times (x)+(a)\times (5)+(3b)\]\[\times (x)+(3b)\,\times (5)\] \[=ax+5a+3bx+15b\] (v) \[(2pq+3{{q}^{2}})\] and \[(3pq-2{{q}^{2}})\] \[(2pq+3{{q}^{2}})\times (3pq-2{{q}^{2}})\] \[=(2pq)\times (3pq-2{{p}^{2}})+(3{{p}^{2}})\]\[\times (3pq-2{{q}^{2}})\] \[=(2pq)\times (3pq)-(2pq)\times (2{{q}^{2}})\] \[+(3{{p}^{2}})\times (3pq)-(3{{q}^{2}})\times (2{{q}^{2}})\] \[=6{{p}^{2}}{{q}^{2}}-4p{{q}^{3}}+9p{{q}^{3}}-6{{q}^{4}}\] \[=6{{p}^{2}}{{q}^{2}}\,+(9p{{q}^{3}}-4p{{q}^{3}})-6{{q}^{4}}\]                  |Combining like terms \[=6{{p}^{2}}{{q}^{2}}+5p{{q}^{3}}-6{{q}^{4}}\] (vi) \[\left( \frac{3}{4}{{a}^{2}}+3{{b}^{2}} \right)\] and \[4\left( {{a}^{2}}-\frac{2}{3}{{b}^{2}} \right)\] \[\left( \frac{3}{4}{{a}^{2}}+3{{b}^{2}} \right)\times 4\left( {{a}^{2}}-\frac{2}{3}{{b}^{2}} \right)\] \[=\left( \frac{3}{4}{{a}^{2}}+3{{b}^{2}} \right)\,\times \left( 4{{a}^{2}}-\frac{8}{3}{{b}^{2}} \right)\] \[=\frac{3}{4}{{a}^{2}}\times \left( 4{{a}^{2}}-\frac{8}{3}{{b}^{2}} \right)+3{{b}^{2}}\]\[\times \,\left( 4{{a}^{2}}-\frac{8}{3}{{b}^{2}} \right)\]                 \[=\left( \frac{3}{4}{{a}^{2}} \right)\times \,(4{{a}^{2}})-\left( \frac{3}{4}{{a}^{2}} \right)\times \left( \frac{8}{3}{{b}^{2}} \right)\] \[+(3{{b}^{2}})\times \,(4{{a}^{2}})-(3{{b}^{2}})\times \left( \frac{8}{3}{{b}^{2}} \right)\] \[=3{{a}^{4}}-2{{a}^{2}}{{b}^{2}}+12{{b}^{2}}{{a}^{2}}-8{{b}^{4}}\] \[=3{{a}^{4}}+(12{{a}^{2}}{{b}^{2}}-2{{a}^{2}}{{b}^{2}})-8{{b}^{4}}\]        |Combining like terms \[=3{{a}^{4}}+10{{a}^{2}}{{b}^{2}}-8{{b}^{4}}\].