question_answer 1)

                Factorize the expressions:                 (i) \[a{{x}^{2}}+bx\]                                        (ii) \[7{{p}^{2}}+21{{q}^{2}}\]                     (iii) \[2{{x}^{3}}+2x{{y}^{2}}+2x{{z}^{2}}\]                 (iv) \[a{{m}^{2}}+b{{m}^{2}}+b{{n}^{2}}+a{{n}^{2}}\]      (v) \[(lm+l)\,+m+l\]        (vi) \[y(y+z)\,+9\,(y+z)\]                 (vii) \[5{{y}^{2}}-20y-8z+2yz\]    (viii) \[10ab+4a+5b+2\] (ix) \[6xy-4y+6-9x\].

Answer:

                (i) \[a{{x}^{2}}+bx\]                        \[a{{x}^{2}}+bx=x(ax+b)\] (ii) \[7{{p}^{2}}+21{{q}^{2}}\] \[7{{p}^{2}}+21{{q}^{2}}\,=7({{p}^{2}}+3{{q}^{2}})\]                 (iii) \[2{{x}^{3}}+2x{{y}^{2}}+2x{{z}^{2}}\]             \[2{{x}^{3}}+2x{{y}^{2}}+2x{{z}^{2}}\]\[=2x({{x}^{2}}+{{y}^{2}}+{{z}^{2}})\]                 (iv) \[a{{m}^{2}}+b{{m}^{2}}+b{{n}^{2}}+a{{n}^{2}}\]                      \[a{{m}^{2}}+b{{m}^{2}}+b{{n}^{2}}+a{{n}^{2}}\] \[=a{{m}^{2}}+b{{m}^{2}}+a{{n}^{2}}+b{{n}^{2}}\] \[={{m}^{2}}(a+b)\,+{{n}^{2}}\,(a+b)\] \[=(a+b)\,({{m}^{2}}+{{n}^{2}})\]                 (v) \[(lm+l)\,+m+l\] \[(lm+l)\,+m+l\] \[=l(m+1)\,+1\,(m+1)\] \[=\,(m+1)\,(l+1)\]                 (vi) \[y(y+z)\,+9\,(y+z)\]                 \[y(y+z)\,+9\,(y+z)\]\[=(y+z)\,(y+9)\]                 (vii) \[5{{y}^{2}}-20y-8z+2yz\]    \[5{{y}^{2}}-20y-8z+2yz\] \[=5{{y}^{2}}-20y+2yz-8z\] \[=5y\,(y-4)\,+2z\,(y-4)\] \[=(y-4)\,\,(5y+2z)\]                 (viii) \[10ab+4a+5b+2\]                 \[10ab+4a+5b+2\] \[=2a\,(5b+2)\,+1\,(5b+2)\] \[=(5b+2)\,(2a+1)\]                 (ix) \[6xy-4y+6-9x\].                       \[6xy-4y+6-9x\] \[=6xy-4y-9x+6\] \[=2y\,(3x-2)\,-3(3x-2)\] \[=(3x-2)\,(2y-3)\].