question_answer 16)

                 Find the product:                 (i) \[(5-2x)\,(3+x)\,\]      (ii) \[(x+7y)\,(7x-y)\]      (iii) \[({{a}^{2}}+b)(a+{{b}^{2}})\]                 (iv) \[({{p}^{2}}-{{q}^{2}})\,(2p+q)\]

Answer:

                (i) \[(5-2x)\,(3+x)\]                 \[(5-2x)\,(3+x)\] \[=(5)\times (3+x)-(2x)\times (3+x)\] \[=(5)\times \left( 3 \right)\,+(5)\,\times (x)\,-(2x)\]\[\times (3)\,-(2x)\times (x)\] \[=15+5x-6x-2{{x}^{2}}\] \[=15-x-2{{x}^{2}}\]                                        |Combining like terms (ii) \[(x+7y)\,(7x-y)\] \[=(x+7y)\,\times (7x-y)\] \[=(x)\times (7x-y)+(7y)\,\times (7x-y)\] \[=(x)\times (7x)-(x)\times (y)\,+(7y)\]\[\times (7x)-(7y)\times (y)\]                 \[=7{{x}^{2}}-xy+49yx-7{{y}^{2}}\] \[=7{{x}^{2}}+48xy-7{{y}^{2}}\]                 |Combining like terms (iii) \[({{a}^{2}}+b)\,(a+{{b}^{2}})\] \[({{a}^{2}}+b)\,(a+{{b}^{2}})\] \[={{a}^{2}}\times \,(a+{{b}^{2}})\,+b\times \,(a+{{b}^{2}})\] \[=({{a}^{2}})\times (a)+({{a}^{2}})\times ({{b}^{2}})+(b)\]\[\times (a)+(b)\times ({{b}^{2}})\] \[={{a}^{3}}+{{a}^{2}}{{b}^{2}}+ba+{{b}^{3}}\] (iv) \[({{p}^{2}}-{{q}^{2}})\,(2p+q)\] \[({{p}^{2}}-{{q}^{2}})\,\times (2p+q)\] \[={{p}^{2}}\times \,(2p+q)-{{q}^{2}}\times \,(2p+q)\] \[=({{p}^{2}})\,\times (2p)+({{p}^{2}})\times (q)-({{q}^{2}})\]\[\times (2p)\,-({{q}^{2}})\times (q)\] \[=2{{p}^{3}}+{{p}^{2}}q-2{{q}^{2}}p-{{q}^{3}}\]