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C2: Chapter 1 Algebra and Functions Dr J Frost ([email protected]) Last modified: 2nd September 2013 Terminology 11 ÷ 4 = 2 3 ? dividend ? divisor quotient ? ? remainder Normal Long Division 38. 11 4 2 3 . 0 0 0 0 33 93 88 5 0 1. We found how many whole number of times (i.e. the quotient) the divisor went into the dividend. 2. We multiplied the quotient by the dividend. 3. …in order to find the remainder. 4. Find we ‘brought down’ the next number. 2 6x x+5 - 2x + 3 6x3 + 28x2 – 7x + 15 3 2 6x + 30x 2 – 2x – 7x – 2x2 – 10x 3x + 15 3x + 15 0 The Anti-Idiot Test: You can check your solution by finding (x+5)(6x2 – 2x + 3) 2 3x x-1 + 0 –4 3x3 – 3x2 – 4x + 4 3 2 3x – 3x 0 – 4x + 4 – 4x + 4 0 2 2x + 3x – 4 x - 4 2x3 – 5x2 – 16x + 10 3 2 2x – 8x Find the 2 3x – 16x remainder. 3x2 – 12x -4x + 10 Q: Is (x-4) a factor of -4x + 16 2x3 – 5x2 – 16x + 10? -6 Exercises 1a 1i 2a 2i 2b Exercise 1B Divide 3 + 6 2 + 8 + 3 by + 1 ? +3 2 + 5 Divide 3 − 8 2 + 13 + 10 by − 5 ? −2 2 − 3 Divide 6 3 + 27 2 + 14 + 8 by + 4 6 2 + 3 ? +2 Divide −5 3 − 27 2 + 23 + 30 by + 6 −5 2 +? 3 + 5 Exercise 1C Find the remainder when 3 + 4 2 − 3 + 2 is divided by + 5. −8 ? Dividing polynomials with ‘missing’ terms Divide x3 – 1 by x – 1 How would we write the division? 2 : + ? + 1 For Olympiad enthusiasts: In general, the difference of two cubes can be factorised as: 3 − 3 = − 2 + + 2 Dividing polynomials with ‘missing’ terms Divide x4 – 16 by (x+2) : 3 − 2 2 ?+ 4 − 8 Recap dividend quotient 8 = 2+ 3 divisor 1 3 remainder Remainder and Factor Theorem We’re trying to work out the remainder when we divide a polynomial by − = + − − () = ( − )() + So what does f(a) equal? What if = 0? Remainder and Factor Theorem ! Remainder Theorem For a polynomial (), the remainder when () is divided by − is . ! Factor Theorem If = 0, then by above, the remainder is 0. Thus ( − ) is a factor of . Basic Examples Remainder when 2 + 1 is divided by − 2? 2 = ?5 Remainder when 3 − is divided by + 1? −1 =?0 Remainder when 2 + 1 is divided by 2 − 1? 1 5 =? 2 4 Remainder when 2 − is divided by 3 + 4? 4 28 − =? 3 9 Examples Show that (x – 2) is a factor of x3 + x2 – 4x - 4 2 = 8 + 4? − 8 − 4 = 0 Examples Fully factorise 2x3 + x2 – 18x – 9 = ( − 3)( +?3)(2 + 1) Tip: If f(x) = 2x3 + x2 – 18x – 9, then try f(-1), f(1), f(2), etc. until one of these is equal to 0. Examples Fully factorise 3 + 6 2 + 5 − 12 = ( − 1)( +? 3)( + 4) Given that ( + 1) is a factor of 44 − 32 + , find the value of . = −1? Examples C2 May 2013 (Retracted) = ,? = − ( − )( − ? )( + ) Examples Exercise 1D Q1, 2, 4, 6, 8, 10 Recap Q10) Given that ( − 1) and ( + 1) are factors of 3 + 2 − 3 − 7 find the value of and . = 3,? = 7 Recap Find the remainder when 16x5 – 20x4 + 8 is divided by (2 − 1) 15 ? 2 Bro tip: think what you could make x in order to make the factor (2x-1) zero. Recap When 84 − 43 + 2 − 1 is divided by (2 + 1) the remainder is 3. Find the value of . =? 16 Exercises Le Exercise 1E: • Q1f, g, h, i • 2, 4, 6, 8, 10 Le Exercise 1F • 4, 5, 8, 10, 15.