question_answer 4)

                Factorize the following expressions:                 (i) \[{{a}^{2}}+8a+16\]                                                                   (ii) \[{{p}^{2}}-10p+25\]                 (iii) \[25{{m}^{2}}+30m+9\]                                                         (iv) \[49{{y}^{2}}+84yz+36{{z}^{2}}\]                 (v) \[4{{x}^{2}}-8x+4\]                                                                   (vi) \[121{{b}^{2}}-88bc+16{{c}^{2}}\]                 (vii) \[{{(l+m)}^{2}}-4lm\]                                                             (Hint: Expand \[{{(l+m)}^{2}}\] first)                 (viii) \[{{a}^{4}}+2{{a}^{2}}{{b}^{2}}+{{b}^{4}}\].               

Answer:

                (i) \[{{a}^{2}}+8a+16\]                 \[{{a}^{2}}+8a+16\]\[={{(a)}^{2}}+2(a)(4)+{{(4)}^{2}}\] \[={{(a+4)}^{2}}\]                                                            |Applying Identity I                 (ii) \[{{p}^{2}}-10p+25\] \[{{p}^{2}}-10p+25\] \[={{(p)}^{2}}-2(p)\,(5)\,+{{(5)}^{2}}\] \[={{(p-5)}^{2}}\]                                                             |Using Identity II                 (iii) \[25{{m}^{2}}+30m+9\]                         \[25{{m}^{2}}+30m+9\] \[={{(5m)}^{2}}\,+2(5m)\,(3)+{{(3)}^{2}}\] \[={{(5m+3)}^{2}}\]                                                        |Applying Identity I (iv) \[49{{y}^{2}}+84yz+36{{z}^{2}}\] \[49{{y}^{2}}+84yz+36{{z}^{2}}\] \[={{(7y)}^{2}}+2(7y)\,(6z)\,+{{(6z)}^{2}}\] \[={{(7y+6z)}^{2}}\]                                                        |Using Identity I                 (v) \[4{{x}^{2}}-8x+4\]   \[4{{x}^{2}}-8x+4\] \[=4({{x}^{2}}-2x+1)\] \[=4[{{(x)}^{2}}-2(x)(1)+{{(1)}^{2}}]\] \[=4{{(x-1)}^{2}}\]                                                           |Applying Identity II                 (vi) \[121{{b}^{2}}-88bc+16{{c}^{2}}\]     \[121{{b}^{2}}-88bc+16{{c}^{2}}\] \[={{(11b)}^{2}}\,-2(11b)\,(4c)\,+{{(4c)}^{2}}\] \[={{(11b-4c)}^{2}}\]                                                      |Using Identity II                 (vii) \[{{(l+m)}^{2}}-4lm\]                 \[{{(l+m)}^{2}}-4lm\] \[=({{l}^{2}}+2lm+{{m}^{2}})\,-4lm\]                                       |Using Identity I \[={{l}^{2}}+(2lm-4lm)+{{m}^{2}}\]                                          |Combining the like terms \[={{l}^{2}}-2lm+{{m}^{2}}\] \[={{(l)}^{2}}-2(l)(m)+{{(m)}^{2}}\] \[={{(l-m)}^{2}}\]                                                             |Applying Identity II                 (viii) \[{{a}^{4}}+2{{a}^{2}}{{b}^{2}}+{{b}^{4}}\]                 \[{{a}^{4}}+2{{a}^{2}}{{b}^{2}}+{{b}^{4}}\] \[={{({{a}^{2}})}^{2}}+2({{a}^{2}})\,({{b}^{2}})+{{({{b}^{2}})}^{2}}\] \[={{({{a}^{2}}+{{b}^{2}})}^{2}}\]                                                              |Using Identity I