question_answer 8)

                Factorize the following expressions.                 (i) \[{{p}^{2}}+6p+8\]                     (ii) \[{{q}^{2}}-10q+21\]                (iii) \[{{p}^{2}}+6p-16\]

Answer:

                (i) \[{{p}^{2}}+6p+8\]     \[{{p}^{2}}+6p+8\]                          \[={{p}^{2}}+6p+9-1\] \[=\{{{(p)}^{2}}+2\,(p)\,(3)\,+{{(3)}^{2}}\}\,-{{(1)}^{2}}\] \[={{(p+3)}^{2}}-{{(1)}^{2}}\]                                      |Using Identity I \[=(p+3-1)\,(p+3+1)\]                    |Using Identity III \[=(p+2)\,(p\,+4)\] (ii) \[{{q}^{2}}-10q+21\] \[{{q}^{2}}-10q+21\] \[={{q}^{2}}-10q+25-4\] \[=\{{{(p)}^{2}}-2(q)\,(5)\,+{{(5)}^{2}}\}-4\]        \[={{(q-5)}^{2}}\,-{{(2)}^{2}}\]                                   |Using Identity II \[=(q-5-2)\,(q-5+2)\]                      |Using Identity III \[=(q-7)\,(q-3)\] (iii) \[{{p}^{2}}+6p-16\]                  \[{{p}^{2}}+6p-16\]  \[={{p}^{2}}+6p+9-25\] \[={{(p)}^{2}}+2(p)\,(3)+{{(3)}^{2}}-{{(5)}^{2}}\] \[={{(p+3)}^{2}}-{{(5)}^{2}}\]                      |Using Identity I \[=(p+3-5)\,(p+3+5)\]                    |Applying Identity III \[=(p-2)\,(p+8)\]