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Section 2.4
Dividing Polynomials;
Remainder and Factor Theorems
Long Division of Polynomials
and
The Division Algorithm
Long Division of Polynomials
Long Division of Polynomials
3x  4
3x  2 9 x  6 x  5
2
9x  6x
12 x  5
2
13

3x  2
12 x  8
13
Long Division of Polynomials with Missing Terms
x 5
2
x +5x -3 x
31x  17
 2
x  5x  3
 3x  2
3
x 3 +5x 2  3x
-5x  6x  2
2
-5x 2  25x  15
31x- 17
You need to leave a hole when you have
missing terms. This technique will help
you line up like terms. See the dividend
above.
Example
Divide using Long Division.
2 x  5 6 x  3x
3
2
7
Example
Divide using Long Division.
x 2  2 x  1 8 x 4  3x3
+5x  1
Dividing Polynomials Using
Synthetic Division
Comparison of Long Division and Synthetic
Division of X3 +4x2-5x+5 divided by x-3
Steps of Synthetic Division dividing 5x3+6x+8 by x+2
Put in a 0 for the missing term.
Using synthetic division instead of long division.
Notice that the divisor has to be a binomial of
degree 1 with no coefficients.
2 5 + 7 - 1
10 6
5 3 5
Thus:
5
5x  3 
x2
x  2 5x2  7 x 1
Example
Divide using synthetic division.
3x3  5 x 2  7 x  8
x4
The Remainder Theorem
If you are given the function f(x)=x3- 4x2+5x+3 and
you want to find f(2), then the remainder of this
function when divided by x-2 will give you f(2)
f(2)=5
f (1) for f(x)=6x 2  2 x  5 is
1 6
-2 5
6
6 4
4 9
f(1)=9
Example
Use synthetic division and the remainder
theorem to find the indicated function value.
f ( x)  3x3  5x 2  1; f(2)
The Factor Theorem
Solve the equation 2x3-3x2-11x+6=0 given that 3 is
a zero of f(x)=2x3-3x2-11x+6. The factor theorem
tells us that x-3 is a factor of f(x). So we will use
both synthetic division and long division to show
this and to find another factor.
Another factor
Example
Solve the equation 5x2 + 9x – 2=0 given
that -2 is a zero of f(x)= 5x2 + 9x - 2
Example
Solve the equation x3- 5x2 + 9x - 45 = 0 given
that 5 is a zero of f(x)= x3- 5x2 + 9x – 45.
Consider all complex number solutions.
Divide
x
3
 x  2 x  8    x  3
2
2
x
 x 8
(a)
2
x
 4x  2
(b)
2
x
 4 x  14
(c)
34
(d) x  4 x  14 
x 3
2
Use Synthetic Division and the Remainder
Theorem to find the value of f(2) for the function
f(x)=x 3 +x 2 - 11x+10
(a) 2
(b) 0
(c) 5
(d) 12