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Find the common factors of the given terms: (i) \[12x,\,36\] (ii) \[2y,\,22xy\] (iii) \[14pq,\,28{{p}^{2}}{{q}^{2}}\] (iv) \[2x,\,3{{x}^{2}},4\] (v) \[6abc,\,24a{{b}^{2}},\,12{{a}^{2}}b\] (vi) \[16{{x}^{3}},\,-4{{x}^{2}},\,32x\] (vii) \[10pq,\,20qr,\,30rp\] (viii) \[3{{x}^{2}}{{y}^{3}},\,10{{x}^{3}}{{y}^{2}},\,6{{x}^{2}}{{y}^{2}}z\].
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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) \[a{{x}^{2}}y\,+bx{{y}^{2}}+cxyz\].
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Factorize: (i) \[{{x}^{2}}+xy+8x+8y\] (ii) \[15xy-6x+5y-2\] (iii) \[ax+bx-ay-by\] (iv) \[15pq+15+9q+25p\] (v) \[z-7+7xy-xyz\].
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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) \[{{a}^{4}}+2{{a}^{2}}{{b}^{2}}+{{b}^{4}}\].
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Factorize: (i) \[4{{p}^{2}}-9{{q}^{2}}\] (ii) \[63{{a}^{2}}-112{{b}^{2}}\] (iii) \[49{{x}^{2}}-36\] (iv) \[16{{x}^{5}}-144{{x}^{3}}\] (v) \[{{(l+m)}^{2}}-{{(l-m)}^{2}}\] (vi) \[9{{x}^{2}}{{y}^{2}}-16\] (vii) \[({{x}^{2}}-2xy+{{y}^{2}})-{{z}^{2}}\] (viii) \[25{{a}^{2}}-4{{b}^{2}}+28bc-49{{c}^{2}}\].
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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) \[6xy-4y+6-9x\].
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Factorize: (i) \[{{a}^{4}}-{{b}^{4}}\] (ii) \[{{p}^{4}}-81\] (iii) \[{{x}^{4}}-{{(y+z)}^{4}}\] (iv) \[{{x}^{4}}-{{(x-z)}^{4}}\] (v) \[{{a}^{4}}-2{{a}^{2}}{{b}^{2}}+{{b}^{4}}\]
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Factorize the following expressions. (i) \[{{p}^{2}}+6p+8\] (ii) \[{{q}^{2}}-10q+21\] (iii) \[{{p}^{2}}+6p-16\]
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Carry out the following divisions: (i) \[28{{x}^{4}}\div 56x\] (ii) \[-36{{y}^{3}}\div 9{{y}^{2}}\] (iii) \[66p{{q}^{2}}{{r}^{3}}\div \,11q{{r}^{2}}\] (iv) \[34{{x}^{3}}{{y}^{3}}{{z}^{3}}\div \,51x{{y}^{2}}{{z}^{3}}\] (v) \[12{{a}^{8}}{{b}^{8}}\div \,(-6{{a}^{6}}{{b}^{4}})\]
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Divide the given polynomial by the given monomial. (i) \[(5{{x}^{2}}-6x)\div 3x\] (ii) \[(3{{y}^{8}}-4{{y}^{6}}+5{{y}^{4}})\div {{y}^{4}}\] (iii) \[8({{x}^{3}}{{y}^{2}}{{z}^{2}}+{{x}^{2}}{{y}^{3}}{{z}^{2}}+{{x}^{2}}{{y}^{2}}{{z}^{3}})\div 4{{x}^{2}}{{y}^{2}}{{z}^{2}}\] (iv) \[({{x}^{3}}+2{{x}^{2}}+3x)\,\div 2x\] (v) \[({{p}^{3}}{{q}^{6}}-{{p}^{6}}{{q}^{3}})\div \,{{p}^{3}}{{q}^{3}}\].
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Work out the following divisions: (i) \[(10x-25)\div 5\] (ii) \[(10x-25)\div \,(2x-5)\] (iii) \[10y(6y+21)\,\div 5(2y+7)\] (iv) \[9{{x}^{2}}{{y}^{2}}\,(3z-24)\div \,27xy\,(z-8)\] (v) \[96abc(3a-12)\,(5b-30)\div \,144(a-4)\,(b-b)\].
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Divide as directed. (i) \[5(2x+1)\,(3x+5)\,\div (2x+1)\] (ii) \[26xy(x+5)\,(y-4)\,\div \,13x(y-4)\] (iii) \[52pqr\,(p+q)\,(q+r)\,(r+p)\,\div \,104pq\] \[(q+r)\,(r+p)\] (iv) \[20(y+4)({{y}^{2}}+5y+3)\,\div \,5(y+4)\] (v) \[x(x+1)\,(x+2)\,(x+3)\,\div x(x+1)\].
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Factorize the expressions and divide them as directed. (i) \[({{y}^{2}}+7y+10)\div (y+5)\] (ii) \[({{m}^{2}}-14m-32)\div (m+2)\] (iii) \[(5{{p}^{2}}-25p+20)\div (p-1)\] (iv) \[4yz({{z}^{2}}+6z-16)\div \,2y(z+8)\] (v) \[5pq({{p}^{2}}-{{q}^{2}})\,\div 2p\,(p+q)\] (vi) \[12xy\,(9{{x}^{2}}-16{{y}^{2}})\div 4xy\,(3x+4y)\] (vii) \[39{{y}^{3}}(50{{y}^{2}}-98)\,\div 26{{y}^{2}}(5y+7)\].
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Find and correct the errors in the following mathematical statements. 1. \[4(x-5)\,=4x-5\] 2. \[x(3x+2)\,=3{{x}^{2}}+2\] 3. \[2x+3y=5xy\] 4. \[x+2x+3x=5x\] 5. \[5y+2y+y-7y=0\] 6. \[3x+2x=5{{x}^{2}}\] 7. \[{{(2x)}^{2}}+4(2x)+7=2{{x}^{2}}+8x+7\] 8. \[{{(2x)}^{2}}+5x=4x+5x=9x\] 9. \[{{(3x+2)}^{2}}=3{{x}^{2}}+6x+4\]. 10. Substituting \[x=-3\] in (a) \[{{x}^{2}}+5x+4\] gives \[{{(-3)}^{2}}\,+5(-3)\,+4\] \[=9+2+4=15\] (b) \[{{x}^{2}}-5x+4\] gives \[{{(-3)}^{2}}-5(-3)+4\] \[=9-15+4=-2\] (c) \[{{x}^{2}}+5x\] gives \[{{(-3)}^{2}}\,+5(-3)\] \[=-9-15=-24\]. 11. \[{{(y-3)}^{2}}={{y}^{2}}-9\] 12. \[{{(z+5)}^{2}}={{z}^{2}}+25\] 13. \[(2a+3b)\,(a-b)\,=2{{a}^{2}}-3{{b}^{2}}\] 14. \[(a+4)\,(a+2)\,={{a}^{2}}+8\] 15. \[(a-4)\,(a-2)\,={{a}^{2}}-8\] 16. \[\frac{3{{x}^{2}}}{3{{x}^{2}}}=0\] 17. \[\frac{3{{x}^{2}}+1}{3{{x}^{2}}}=1+1=2\] 18. \[\frac{3x}{3x+2}\,=\frac{1}{2}\] 19. \[\frac{3}{4x+3}\,=\frac{1}{4x}\] 20. \[\frac{4x+5}{4x}=5\] 21. \[\frac{7x+5}{5}=7x\]
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