Synthetic Division

1
Synthetic Division
Find the remainder of f(x) when divided by x-a.
Ex Find the remainder of x4 2x3 - 6x27x-13
when divided by x-2.
Method 1 Using long division
Method 2 Using Synthetic Division
2
How synthetic division works

1. 1 2 -6 7 -13

2x1
add
2
8
4
22
_________________________________
1
4
2
11
9
The quotient is x34x22x11.
The remainder is 9.
3
Exercise

• Find the remainder when 4x3 2x-5
• is divided by
• x 3
• x 5

Finish both parts before you go to the next slide.
4
Solution Remember that coefficient of x2 0.
(a) 3 4 0 2 -5
12
36
114
_____________________________
4
12
38
109
Remainder 28
Write x5 as x-(-5)
(b) -5 4 0 2 -5
-510
-20
100
____________________________
-515
4
-20
102
Remainder -515
5
Variation
From previous example, 4x32x-5 when divided by
x-3 leaves a quotient 4x212x38 and the
remainder is 109.
Exercise Find the quotient and the remainder
when 4x32x-5 is divided by 2x-6.
6
Solution
Since 4x32x-5 when divided by x-3 leaves a
quotient 4x212x38 and a remainder 109,
we can write 4x32x-5(x-3)(4x212x38)109.
To find the quotient and the remainder when
4x32x-5 is divided by 2x-6.
4x32x-5(x - 3)(4x2 12x38)109
(2x-6)(2x2 6x19)109
Thus, the quotient is 2x26x19 and the
remainder is 109.
7
Last exercise
Find the quotient and remainder when
f(x)4x32x2-3x1 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.