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Introduction, Factors of natural numbers, e.g. 90 = 2 x 3 x 3 x 5, Factors of algebraic expressions, Terms are formed as product of factors., e.g. 5xy = 5 x x x y, 10x(x+2)(y+3)=2x5xxx(x+2)x(y+3)

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Method of common factors, , • We write each term as a product of, irreducible factors. Then take out the, common factors of the terms and write the, remaining factors to get the desired factor, form., • E.g. 5ab+10a, (5xaxb)+(10xa), (5axb) + (5ax2), 5a( b+2 ) (desired factor form)

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Let us solve some examples

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Factorise, 2, 2, 2, a bc + ab c + abc, Take the common factors, abc ( a + b +c ), a x b x c x (a + b + c)

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Factorisation using identities, , • Observe the expression., , • If it has a form that fits the right hand side of one of the identities ,, then the expression corresponding to the left hand side of the, identity gives the desired factorisation., • (a+b)2 = a2+2ab+b2, • (a-b)2 = a2 -2ab+b2, • (a2-b2) = (a+b)(a-b)

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Factors of the form (x+a)(x+b), • In general , for factorising an algebraic expression of the, type x2+px+q , we find two factors a and b of q (i.e. the, constant term) such that, ab = q, a+b = p

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a  2a b  b, 4, , 2 2, , 4, , (a )  2a b  (b ), 2 2, , 2 2, , (a  b ), 2, , 2 2, , 2 2

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4x  8x  4, 2, , 4( x  2 x  1), 2, , 4( x  x  x  1), 2, , 4[ x( x  1)  1( x  1)], 4( x  1)( x  1), 4( x  1), , 2

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p 6p 8, 2, , p  4p  2p 8, p ( p  4)  2( p  4), 2, , ( p  2)( p  4)

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a  10a  21, 2, , a  7 a  3a  21, 2, , a (a  7)  3( a  7), (a  7)( a  3)

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Division of algebraic expressions, • Division of monomial by another monomial, • Division of polynomial by a monomial, • Division of polynomial by polynomial

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Division of monomial by another, monomial, 6 x3  2 x, Now let us write the irreducible factor forms, , 6 x3  2  3  x  x  x, 2x  2  x,  6 x 3  2  x  (3  x  x ),  (2 x )  (3 x 2 ),  6 x3  2 x  3x 2

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Division of polynomial by a, monomial, , (5 x  6 x )  3 x, x (5 x  6), 3 x, (5 x  6), 3, 2

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( x  2 x  3x)  2 x, 3, , 2, , x( x  2 x  3), 2 x, 2, ( x  2 x  3), 2, 2

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Division of polynomial by, polynomial, ( y  7 y  10)  ( y  5), 2, , ( y  5 y  2 y  10), ( y  5), [ y ( y  5)  2( y  5)], ( y  5), ( y  5)( y  2), ( y  5), ( y  2), 2

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( m  14m  32)  ( m  2), 2, , ( m  16m  2m  32), ( m  2), [ m( m  16)  2( m  16)], ( m  2), ( m  16)( m  2), ( m  2), ( m  16), 2

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CONCEPT, MAP