2. Synthetic Division Find the remainder of f(x) when divided by x-a. ExFind the remainder of x 4 +2x 3 - 6x 2 +7x-13 when divided by x-2. Method 1Using long division Method 2 Using Synthetic Division

3. 2| 12-67-13 _________________________________ 1 2 42119 8422 The quotient is x 3 +4x 2 +2x+11. The remainder is 9. How synthetic division works: 2x1 add

4. Exercise Find the remainder when 4x 3 +2x-5 is divided by (a)x – 3 (b)x + 5 Finish both parts before you go to the next slide.

5. Solution Remember that coefficient of x 2 = 0. (a)3 | 4 0 2 -5 (b)-5|4 0 2 -5 Write x+5 as x-(-5) _____________________________ 4 12 36 38 114 109 Remainder = 28 ____________________________ 4 -20 100 102 -510 -515 Remainder = -515

6. Variation From previous example, 4x 3 +2x-5 when divided by x-3 leaves a quotient 4x 2 +12x+38 and the remainder is 109. Exercise Find the quotient and the remainder when 4x 3 +2x-5 is divided by 2 x-6.

7. Solution Since 4x 3 +2x-5 when divided by x-3 leaves a quotient 4x 2 +12x+38 and a remainder 109, we can write 4x 3 +2x-5=(x-3)(4x 2 +12x+38)+109. To find the quotient and the remainder when 4x 3 +2x-5 is divided by 2x-6. 4x 3 +2x-5=(x - 3)(4x 2 + 12x+38)+109 Thus, the quotient is 2x 2 +6x+19 and the remainder is 109. =(2x-6)(2x 2 + 6x+19)+109

8. Last exercise Find the quotient and remainder when f(x)=4x 3 +2x 2 -3x+1 is divided by 2x-1. Hint: Write 2x-1 = 2(x-1/2) Apply synthetic division using x-1/2 Deduce the quotient and remainder when f(x) is divided by 2x-1. We shall check the answer in class.