Question 1: Find the following products;

i) $( x+5 ) ( x+7 )$     ii) $( b+2 ) ( b+9 )$      iii) $( c+2 ) ( c+$ $\frac{3}{5}$ $)$      iv) $( t+$ $\frac{4}{3}$ $) ( t+$ $\frac{1}{3}$ $)$

i) $( x+5 ) ( x+7 )$ $= x^2+5x+7x+35$ $= x^2+12x+35$

ii) $( b+2 ) ( b+9 )$ $= b^2+2b+9b+18$ $= b^2+11b+18$

iii) $( c+2 ) ( c+$ $\frac{3}{5}$ $) = c^2+2c+$ $\frac{3}{5}$ $c+$ $\frac{6}{5}$ $= c^2+ ( 2+$ $\frac{3}{5}$ $) c+$ $\frac{6}{5}$ $= c^2+$ $\frac{13}{5}$ $c+$ $\frac{6}{5}$

iv) $( t+$ $\frac{4}{3}$ $) ( t+$ $\frac{1}{3}$ $) = { t}^2+$ $\frac{4}{3}$ $t+$ $\frac{1}{3}$ $t+$ $\frac{4}{9}$ $= { t}^2+$ $\frac{5}{3}$ $t+$ $\frac{4}{9}$

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Question 2: Find the following products:

i) $( y+8 ) ( y-4 )$     ii) $( z+6 ) ( z-11 )$    iii) $( c-5 ) ( c+1 )$    iv) $( b-13 ) ( b+10 )$

i) $( y+8 ) ( y-4 ) = y^2+8y-4y-32 = { y}^2+4y-32$

ii) $( z+6 ) ( z-11 ) = { z}^2+6z-11z-66 = { z}^2-5z-66$

iii) $( c-5 ) ( c+1 ) = { c}^2-5c+c-5 = c^2-4c-5$

iv) $( b-13 ) ( b+10 ) = { b}^2-13b+10b-130 = b^2-3b-130$

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Question 3: Find the following products:

i) $( x-3 ) ( x-6 )$    ii) $( z-11 ) ( z-4 )$     iii) $( b-6 ) ( b-8 )$    iv) $( a-$ $\frac{3}{5}$ $) ( a-$ $\frac{1}{3}$ $)$

i)  $( x-3 ) ( x-6 ) = { x}^2-3x-6x+18 = { x}^2-9x+18$

ii)  $( z-11 ) ( z-4 ) = z^2-15z+44 = z^2-11z-4z+44$

iii)  $( b-6 ) ( b-8 ) = b^2-6b-8b+48 = { b}^2-14b+48$

iv)  $( a-$ $\frac{3}{5}$ $) ( a-$ $\frac{1}{3}$ $) = { a}^2-$ $\frac{3}{5}$ $a-$ $\frac{1}{3}$ $a+$ $\frac{1}{5}$ $= a^2-$ $\frac{14}{15}$ $a+$ $\frac{1}{5}$

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Question 4: Explain the following:

i)   $( x-$ $\frac{1}{2}$ $) ( x+$ $\frac{3}{2}$ $)$    ii)  $( p-3 ) ( p+$ $\frac{1}{2}$ $)$

i)  $( x-$ $\frac{1}{2}$ $) ( x+$ $\frac{3}{2}$ $) = { x}^2-$ $\frac{1}{2}$ $x+$ $\frac{3}{2}$ $x-$ $\frac{3}{4}$ $= { x}^2+x-$ $\frac{3}{4}$

ii)  $( p-3 ) ( p+$ $\frac{1}{2}$ $) = { p}^2-3p-$ $\frac{1}{2}$ $p-$ $\frac{3}{2}$ $= { p}^2-$ $\frac{7}{2}$ $p-$ $\frac{3}{2}$

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Question 5: Explain the following:

i)  $( 4p+3 ) ( 4p+7 )$     ii)  $( 9c+4 ) ( 9c-2 )$     ii)  $( 3a-8 ) ( 3a+2 )$

iv)  $( 5x-2 ) ( 5x-7 )$

i)  $( 4p+3 ) ( 4p+7 ) = { 16p}^2+12p+28p+21 = { 16p}^2+40p+21$

ii)  $( 9c+4 ) ( 9c-2 ) = 81c^2+36c-18c-8 = 81c^2+18c-8$

ii)  $( 3a-8 ) ( 3a+2 ) = {9a}^2-24a+6a-16 = { 9a}^2-18a-16$

iv)  $( 5x-2 ) ( 5x-7 ) = {25x}^2-10x-35x+14 = { 25x}^2-45x+14$

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Question 6: Find the following products:

i)  $( x^2+3 ) ( x^2+6 )$          ii)  $( y^2-1 ) ( y^2+4 )$

iii) $( z^2+3 ) ( z^2-7 )$          iv) $( t^2-2 ) ( t^2-5 )$

i)  $( x^2+3 ) ( x^2+6 ) = { x}^4+{3x}^2+6x^2+18 = x^4+9x^2+18$

ii)  $( y^2-1 ) ( y^2+4 ) = y^4-y^2+{4y}^2-4 = { y}^4+3y^2-4$

iii) $( z^2+3 ) ( z^2-7 ) = { z}^4+3z^2-7z^2-21 = { z}^4-4z^2-21$

iv) $( t^2-2 ) ( t^2-5 ) = { t}^4-2t^2-5t^2+10 = t^4-7t^2+10$

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Question 7: Find the following products:

i)  $( ab-2 ) ( ab+4 )$          ii)  $( 2+xy ) ( 3-xy )$

i)  $( ab-2 ) ( ab+4 ) = { a}^2b^2-2ab++4ab-8 = { a}^2b^2+2ab-8$

ii)  $( 2+xy ) ( 3-xy ) = {6+3xy-2xy-x}^2y^2 = { 6+xy-x}^2y^2$

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Question 8: Find the following products:

i)  $( 3x+4y ) ( 4x+3y )$          ii)  $( 4a-5b ) ( 3a+2b )$

iii)  $( 2y+z ) ( 7z-3y )$           iv)  $( 2m-4n ) ( 4m-3n )$

i)  $( 3x+4y ) ( 4x+3y ) = 12x^2+16xy+9xy+12y^2 = 12x^2+25xy+12y^2$

ii)  $( 4a-5b ) ( 3a+2b ) = 12a^2-15ab+8ab-10b^2 = 12a^2-7ab-10b^2$

iii)  $( 2y+z ) ( 7z-3y ) = 14yz+7z^2-6y^2-3yz = 7z^2+11yz-6y^2$

iv)  $( 2m-4n ) ( 4m-3n ) = 8m^2-16mn-9mn-12n^2 = { m}^2-25mn-12n^2$

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Question 9: Find the following products:

i)  $( 2pq+0.1mn ) ( 0.2pq+3mn )$     ii)  $($ $\frac{4m}{p}$ $-$ $\frac{0.2n}{q}$ $) ($ $\frac{3m}{p}$ $+$ $\frac{0.5}{q}$ $)$

iii) $($ $\frac{3}{4}$ $x-2pq ) ( 3x-$ $\frac{4}{5}$ $pq )$

i)  $( 2pq+0.1mn ) ( 0.2pq+3mn )$

$= 0.4p^2q^2+0.02pqmn+6pqmn+0.3m^2n^2$

$= 0.4p^2q^2+6.02pqmn+0.3m^2n^2$

ii)  $($ $\frac{4m}{p}$ $-$ $\frac{0.2n}{q}$ $) ($ $\frac{3m}{p}$ $+$ $\frac{0.5}{q}$ $)$

$=$ $\frac{12m^2}{p^2}$ $-$ $\frac{0.6mn}{pq}$ $+$ $\frac{2mn}{pq}$ $-$ $\frac{0.1n^2}{q^2}$

$=$ $\frac{12m^2}{p^2}$ $+$ $\frac{1.4mn}{pq}$ $-$ $\frac{0.1n^2}{q^2}$

iii) $($ $\frac{3}{4}$ $x-2pq ) ( 3x-$ $\frac{4}{5}$ $pq )$

$=$ $\frac{9}{4}$ $x^2-6pqx-$ $\frac{3}{5}$ $pqx-$ $\frac{8}{5}$ $p^2q^2$

$=$ $\frac{9}{4}$ $x^2-$ $\frac{33}{5}$ $pqx-$ $\frac{8}{5}$ $p^2q^2$

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Question 10: Find the following products:

i)  $( 2a^2+3b^2 ) ( 3a^2+2b^2 )$        ii)  $( x^2+5y^2 ) ( y^2-2x^2 )$

iii)  $( 3c^2-4d^2 ) ( 4c^2-3d^2 )$

i)  $( 2a^2+3b^2 ) ( 3a^2+2b^2 ) = 4a^4+9a^2b^2+4a^2b^2+9a^2b^2 = 4a^4+13a^2b^2+9a^2b^2$
ii)  $( x^2+5y^2 ) ( y^2-2x^2 ) = { x}^2y^2+5y^4-2x^4-10x^2y^2 = 5y^4-9x^2y^2-2x^4$
iii)  $( 3c^2-4d^2 ) ( 4c^2-3d^2 ) = 12c^4-16c^2d^2-9c^2d^2+12d^4 = 12c^4-25c^2d^2+12d^4$