question_answer 2)

                Factorise the following expressions:                 (i) \[7x-42\]                        (ii) \[6p-12q\]                    (iii) \[7{{a}^{2}}+14a\]                    (iii) \[-16z+20{{z}^{3}}\]                 (v) \[5{{x}^{2}}y-15x{{y}^{2}}\]  (vi) \[5{{x}^{2}}y-15x{{y}^{2}}\] (vii) \[10{{a}^{2}}-15{{b}^{2}}+20{{c}^{2}}\]                (viii) \[-4{{a}^{2}}+4ab-4ca\] (ix) \[{{x}^{2}}yz\,+x{{y}^{2}}z+xy{{z}^{2}}\]       (x) \[a{{x}^{2}}y\,+bx{{y}^{2}}+cxyz\].

Answer:

                (i) \[7x-42\]                        \[7x=7\times x\] \[42=2\times 3\times 7\] \[\therefore \]  \[7x-72=7\times x-2\times 3\times 7\] \[=7\times (x-2\times 3)\] \[-7\times (x-6)\] (ii) \[6p-12q\]    \[6p=2\times 3\times p\] \[12q=2\times 2\times 3\times q\] \[\therefore \]  \[6p-12q=2\times 3\times p-2\times 2\times 3\times q\] \[=2\times 3\times (p-2\times q)\] \[=6(p-2q)\]                 (iii) \[7{{a}^{2}}+14a\]                    \[7{{a}^{2}}=7\times a\times a\] \[14a=2\times 7\times a\] \[\therefore \]  \[7{{a}^{2}}+14a=7\times a\times a+2\times 7\times a\] \[=7\times a\times (a+2)\] \[=7a(a+2)\] (iv) \[-16z+20{{z}^{3}}\] \[16z=2\times 2\times 2\times 2\times z\] \[20{{z}^{3}}=2\times 2\times 5\times z\times z\times z\] \[\therefore \]  \[-16z+20{{z}^{3}}=-2\times 2\times 2\times 2\times z\] \[+2\times 2\times z\times (-2\times 2+5\times z\times z)\] \[=4z\,(-4+5{{z}^{2}})\] (v) \[20{{l}^{2}}m+30alm\] \[20{{l}^{2}}m=2\times 2\times 5\times l\times l\times m\] \[30\,alm\,=2\times 3\times 5\times a\times l\times m\] \[\therefore \]  \[20{{l}^{2}}m\,+30\,alm\,=2\times 2\times 5\times l\times l\] \[\times m+2\times 3\times 5\times a\times l\times m\] \[=2\times 5\times l\times m\times (2\times l+3\times a)\] \[=10lm\,(2l+3a)\] (vi) \[5{{x}^{2}}y-15x{{y}^{2}}\] \[5{{x}^{2}}y=5\times x\times x\times y\] \[15x{{y}^{2}}=3\times 5\times x\times y\times y\] \[\therefore \]  \[5{{x}^{2}}y-15x{{y}^{2}}\,=5\times x\times x\times y-3\times 5\]\[\times x\times y\times y\] \[=5\times x\times y\times (x-3\times y)\] \[=5xy\,(x-3y)\]                 (vii) \[10{{a}^{2}}-15{{b}^{2}}+20{{c}^{2}}\]          \[10{{a}^{2}}=2\times 5\times a\times a\] \[15{{b}^{2}}=3\times 5\times b\times b\] \[20{{c}^{2}}=2\times 2\times 5\times c\times c\] \[\therefore \]  \[10{{a}^{2}}-15{{b}^{2}}+20{{c}^{2}}\,=2\times 5\times a\times a\] \[-3\times 5\times b\times b+2\times 2\times 5\times c\times c\] \[=5\times (2\times a\times a-3\times b\times b+2\times 2\times c\times c)\] \[=5(2{{a}^{2}}-3{{b}^{2}}+4{{c}^{2}})\] (viii) \[-4{{a}^{2}}+4ab-4ca\] \[4{{a}^{2}}=2\times 2\times a\times a\] \[4ab=2\times 2\times a\times b\] \[4ca=2\times 2\times c\times a\] \[\therefore \]  \[-4{{a}^{2}}+4ab\,-4ca=-2\times 2\times a\times a\] \[+2\times 2\,\times a\times b-2\times 2\times c\times a\] \[=2\times 2\times a\times \,(-a+b-c)\] \[=4a\,(-a+b-c)\] (ix) \[{{x}^{2}}yz\,+x{{y}^{2}}z+xy{{z}^{2}}\] \[{{x}^{2}}yz\,=x\times x\times y\times z\] \[x{{y}^{2}}z\,=x\times y\times y\times z\] \[xy{{z}^{2}}\,=x\times y\times z\times z\] \[\therefore \]  \[{{x}^{2}}yz+x{{y}^{2}}z\,+xy{{z}^{2}}\] \[=x\times x\times y\times z+x\times y\times y\]\[\times \,z+x\times y\times z\times z\] \[=x\times y\times z\times (x+y+z)\] \[=xyz\,(x+y+z)\] (x) \[a{{x}^{2}}y\,+bx{{y}^{2}}+cxyz\]. \[a{{x}^{2}}y\,=a\times x\times x\times y\] \[bx{{y}^{2}}=b\times x\times y\times y\] \[cxyz\,=c\times x\times y\times z\] \[\therefore \]  \[a{{x}^{2}}y\,+bx{{y}^{2}}\,+cxyz\] \[=a\times x\times x\times y+b\times x\times y\times y\]\[+c\times x\times y\times z\] \[=x\times y\times (a\times x+b\times y+c\times z)\] \[=xy\,(ax+by+cz)\].