#### Transcript Algebraic Long Division

C2: Chapter 1 Algebra and Functions Dr J Frost ([email protected]) Last modified: 2nd September 2013 Terminology 11 ÷ 4 = 2 πππ 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. β¦in order to find the remainder. 4. Find we βbrought downβ the next number. 2 6x x+5 - 2x + 3 6x3 + 28x2 β 7x + 15 3 2 6x + 30x 2 β 2x β 7x β 2x2 β 10x 3x + 15 3x + 15 0 The Anti-Idiot Test: You can check your solution by finding (x+5)(6x2 β 2x + 3) 2 3x x-1 + 0 β4 3x3 β 3x2 β 4x + 4 3 2 3x β 3x 0 β 4x + 4 β 4x + 4 0 2 2x + 3x β 4 x - 4 2x3 β 5x2 β 16x + 10 3 2 2x β 8x Find the 2 3x β 16x remainder. 3x2 β 12x -4x + 10 Q: Is (x-4) a factor of -4x + 16 2x3 β 5x2 β 16x + 10? -6 Exercises 1a 1i 2a 2i 2b Exercise 1B Divide π₯ 3 + 6π₯ 2 + 8π₯ + 3 by π₯ + 1 ? +3 π₯ 2 + 5π₯ Divide π₯ 3 β 8π₯ 2 + 13π₯ + 10 by π₯ β 5 ? β2 π₯ 2 β 3π₯ Divide 6π₯ 3 + 27π₯ 2 + 14π₯ + 8 by π₯ + 4 6π₯ 2 + 3π₯ ? +2 Divide β5π₯ 3 β 27π₯ 2 + 23π₯ + 30 by π₯ + 6 β5π₯ 2 +? 3π₯ + 5 Exercise 1C Find the remainder when π₯ 3 + 4π₯ 2 β 3π₯ + 2 is divided by π₯ + 5. β8 ? Dividing polynomials with βmissingβ terms Divide x3 β 1 by x β 1 How would we write the division? 2 π΄ππ π€ππ: π₯ + ?π₯ + 1 For Olympiad enthusiasts: In general, the difference of two cubes can be factorised as: π₯ 3 β π¦ 3 = π₯ β π¦ π₯ 2 + π₯π¦ + π¦ 2 Dividing polynomials with βmissingβ terms Divide x4 β 16 by (x+2) π΄ππ π€ππ: π₯ 3 β 2π₯ 2 ?+ 4π₯ β 8 Recap dividend quotient 8 = 2+ 3 divisor 1 3 remainder Remainder and Factor Theorem Weβre trying to work out the remainder when we divide a polynomial π π₯ by π₯ β π π π₯ π =π π₯ + π₯βπ π₯βπ π(π₯) = (π₯ β π)π(π₯) + π So what does f(a) equal? What if π π = 0? Remainder and Factor Theorem ! Remainder Theorem For a polynomial π(π₯), the remainder when π(π₯) is divided by π₯ β π is π π . ! Factor Theorem If π π = 0, then by above, the remainder is 0. Thus (π₯ β π) is a factor of π π₯ . Basic Examples Remainder when π₯ 2 + 1 is divided by π₯ β 2? π 2 = ?5 Remainder when π₯ 3 β π₯ is divided by π₯ + 1? π β1 =?0 Remainder when π₯ 2 + 1 is divided by 2π₯ β 1? 1 5 π =? 2 4 Remainder when π₯ 2 β π₯ is divided by 3π₯ + 4? 4 28 π β =? 3 9 Examples Show that (x β 2) is a factor of x3 + x2 β 4x - 4 π 2 = 8 + 4? β 8 β 4 = 0 Examples Fully factorise 2x3 + x2 β 18x β 9 = (π₯ β 3)(π₯ +?3)(2π₯ + 1) Tip: If f(x) = 2x3 + x2 β 18x β 9, then try f(-1), f(1), f(2), etc. until one of these is equal to 0. Examples Fully factorise π₯ 3 + 6π₯ 2 + 5π₯ β 12 = (π₯ β 1)(π₯ +? 3)(π₯ + 4) Given that (π₯ + 1) is a factor of 4π₯4 β 3π₯2 + π, find the value of π. π = β1? Examples C2 May 2013 (Retracted) π = π,? π = βπ (π β π)(ππ β ? π)(ππ + π) Examples Exercise 1D Q1, 2, 4, 6, 8, 10 Recap Q10) Given that (π₯ β 1) and (π₯ + 1) are factors of ππ₯3 + ππ₯2 β 3π₯ β 7 find the value of π and π. π = 3,?π = 7 Recap Find the remainder when 16x5 β 20x4 + 8 is divided by (2π₯ β 1) 15 π ππππππππ ππ ? 2 Bro tip: think what you could make x in order to make the factor (2x-1) zero. Recap When 8π₯4 β 4π₯3 + ππ₯2 β 1 is divided by (2π₯ + 1) the remainder is 3. Find the value of π. π =? 16 Exercises Le Exercise 1E: β’ Q1f, g, h, i β’ 2, 4, 6, 8, 10 Le Exercise 1F β’ 4, 5, 8, 10, 15.