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Starter Simplify 1) 6a4b3 x 2ab7 = Expand 6) m(2m2 – 5) = 2) (3ab4)2 = 7) -4p2(2p4 + 3) = 3) 12m4n5 6mn5 = 8) -2p(2q – 5)3 = 4) (2a2b)4 x 4a5 = Factorise 9) 24m3e4 – 32e5 = 8 w 5p + 6 5) (3w 2p +1 2 ) = 10) 16m3n – 20m6n8 = HW Theta pg 13 Done? . Factorising: What do you notice? . 1) 3x 2 + 7x + 2 2×3=6 Factors 1,6 2,3 Ex 3.05 3x 2 + 7x + 2 = 3x2 + 1x + 6x + 2 = x(3x + 1) + 2(3x + 1) = (3x + 1)(x + 2) 2) 12 x2 17x 5 -5 × 12 = -60 Factors 1,-60 2,-30 12 x2 3,-20 Which add to -17? 4,-15 17x 5 = 12 x2 + 3x 20 x 5 = 3x(4x + 1) 5(4x + 1) Achieve! = (4x + 1)(3x 5) 5,-12 6,-10 . Factorising: What else do you notice? . 1) x 2 + 8x + 16 = (x + 4)(x + 4) Ex 3.06 = (x + 4)2 2) 4x2 + 20 x + 25 25 × 4 = 100, Factors of 100 = 10,10 = 4x2 + 10 x + 10 x + 25 = 2x(2x + 5) + 2x(2x + 5) = (2x + 5)(2x + 5) = (2x + 5)2 3) x2 4) 81x2 5) x2 + 5x 14 2x2 + 12 x 14 100 49y2 = (x + 10 )(x 10 ) Ex 3.06 = (9x + 7y)(9x 7y) (x + 7)(x 2) x +2 = = ( )( ) 2 x + 7 x 1 2(x 1) Ex 3.07 Achieve! eg 7x2 + 13x – 2 Harder Factorising 7x2 + 13x – 2 Multiply the number in front of x2 and the constant List the factors of this number Which pair add to +13 7 × -2 = -14 Factors of -14 = 1 × -14 2 × -7 (number by the x) Rewrite the question splitting the middle term -1 × 14 -2 × 7 +14 and - 1 7x2 + 13x – 2 7x2 + 14x – 1x – 2 Theta Ex 3.05 Factorise each pair 7x(x + 2) – 1(x + 2) Factorise again (7x – 1)(x + 2) Try these 1) 5x2 + 11x + 2 2) 4x2 + 21x + 5 3) 3x2 – 7x – 6 Perfect squares Factorise these quadratics: 1) x2 – 6x + 9 = 3) 9x8 – 42x4y5 + 49y10 = 2) 4x2 – 12x + 9= y y2 4) x 2 x 4 1 Theta Ex 3.06 Difference between two squares Expand (2x + 3)(2x – 3) = So factorising these is easy extension: x2 – 3y + xy – 9 1) x2 – 16 = 4) 4x2 – 4 = 2) 9x2 – 49y2 = 5) 9x2 – 3) 4x – 121y 1 2 1 6) 9 x – 1 6 y2 = 4 6 = 1 4 = Factorising & fractions Algebra fractions can sometimes be simplified by factorising first 4 x 3 1) 2 x3x 4 2) 3x 15x 3) 5 x x 3 4) x 9 x 2 x 2 x 1 4 x 12 5) 2 6 x 2 9x 6) 3x 5x 3 7) 15x 2 10x 6 x 2 9x 8) 2x 3 x 2 8x 12 9) x 2 3x 18 Factorising and cancelling Factorise as much as possible and cancel 3x 9 1) x 2 5x 6 2x 4 2) x 2 4 2x 4 3) x 2 3x 10 2x 2 3x 2 4) 3x 2 4 x 4 x 3 x 2 5x 6 5) x 2 12x 35 x 5 2a 2 98 a 2 3a 10 a 2 5a 14 6) a 2 25 3a 21 2a 10 HW Theta pg 14 Theta Ex 3.07 Factorising Review Is there a common Factor? eg 4x2 + 12x+ 8 Yes No Are there 4 terms (in two pairs)? eg xy + 3x + 2y + 6 Factorise out the common factor first 4(x2 + 3x + 2) Factorise by parts = x(y + 3) + 2( y + 3) = (x + 2)(y + 3) Yes No Is there a difference between squares? Yes eg 4x4 – 9y2 No Factorise (2x2 – 3y) (2x2 + 3y) What is there a coefficient in front of x2 eg 2x2 + 11x + 5 Not 1 =1 Factorise by basic method eg x2 + 3x + 2 = (x + 2)(x + 1) Harder factorising method 2x2 + 11x + 5 = 2x2 + 10x +1x + 5 = 2x(x + 5) + 1(x + 5) = (2x + 1)(x + 5)