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Section 4.4
Synthetic Division; The Remainder and Factor Theorems
Objective 1: Using the Division Algorithm
If we divide 3x4  2 x2  7 x  4 by x2  1 we get
3x 4  2 x 2  7 x  4
8x  7
 3x 2  x  3  2
.
2
x 1
x 1
We see from this division that the quotient is 3x2  x  3 and the remainder is 8x  7 .
Letting f ( x)  3 x 4  2 x 2  7 x  4 , d ( x)  x 2  1 , q( x)  3x 2  x  3 and r ( x)  8 x  7 then
3x 4  2 x 2  7 x  4
8x  7
 3x 2  x  3  2
is equivalent to the equation
2
x 1
x 1
f ( x)
r ( x)

q ( x)

d ( x)
d ( x)
If we multiply both sides of these equations by the divisor d ( x ) , we get:


x2 1
3x 4  2 x 2  7 x  4
8x  7 2
 x 2  1 3x 2  x  3  2
x  1 which is equivalent to
2
x 1
x 1
f ( x)
r ( x)
d ( x)

d ( x) q ( x)

d ( x) or
d ( x)
d ( x)








3x 4  2 x 2  7 x  4  x 2  1 3x 2  x  3  8 x  7  which is equivalent to
f ( x)  d ( x) q ( x)  r ( x) .
The equation f ( x)  d ( x) q( x)  r ( x) is used to check that the long division was done properly. The original
polynomial should equal the product of the divisor and the quotient plus the remainder. This process is
known as the Division Algorithm.
The Division Algorithm
If f ( x) and d ( x ) are polynomial functions with d ( x)  0 , and the degree of d ( x ) is less than or
equal to the degree of f ( x) , then there exist unique polynomial functions q( x) and r ( x ) such that
f ( x)  d ( x)q ( x)  r ( x)
f ( x)
r ( x)
or equivalently
 q ( x) 
d ( x)
d ( x)
where the remainder r ( x)  0 or is of degree less than the degree of d ( x ) .
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Notice in the previous example the remainder, r ( x ) , was a constant. We know that the remainder must
always be of degree less than the degree of the divisor. Therefore, if a polynomial f ( x) is divided by a first
degree polynomial, then the remainder must be a constant.
Corollary to the Division Algorithm
If a polynomial f ( x) of degree greater than or equal to one is divided by a polynomial d ( x ) where d is of
degree one, then there exists a unique polynomial function q( x) and a constant r such that
f ( x)  d ( x)q( x)  r .
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Objective 2
Using Synthetic Division
If a polynomial f ( x) is divided by d ( x)  x  c , then we can use a “short cut method” called synthetic
division to find the quotient, q( x) , and the remainder, r . To illustrate synthetic division we will do a sideby-side comparison of long division and synthetic division using the polynomial function of the previous
example.
LONG DIVISION
SYNTHETIC DIVISION
4 x3  5 x 2  10 x  25
x2
2 4 3 0 5 6
4 x 4  3x3  0 x 2  5 x  6
8 10 20 50
4 x 4  8 x3
4 5 10 25 44
5 x3  0 x 2
5 x3  10 x 2
10 x 2  5 x
10 x 2  20 x
25 x  6
25 x  50
44
You can see how much more compact the synthetic division process is compared to long division. This is
how it works.
Step 1:
The constant coefficient, c, of the divisor d ( x)  x  c is written to the far left while all
coefficients of the polynomial f ( x) are written inside the symbol
. Once
again, be sure to include the 0 for 0x 2 .
This is c in x  c.
2 4 3 0 5 6
These are the coefficients of f  x   4 x 4  3x3  5 x  6.
Step 2:
Bring down the leading coefficient 4
2 4 3 0 5 6

4
Step 3:
Multiply c times the leading coefficient that was just brought down. In this case we multiply
2  4 . The product (8 in this case) is written in the next column in the second row.
2 4 3 0 5 6
Multiply 2  4=8
 8
4
4
Step 4:
Add down the column and write the sum (5 in this case) in the bottom row.
2 4 3 0 5 6
8
4 5
We now repeat this process multiplying c  2 times the value in the last row, always adding down the
columns.
Mulitiply 2  5  10
2 4 3 0 5 6
8 10
4 5
Mulitiply 2  25  50
Add 0+10=10
2 4 3 0 5 6
Mulitiply 2  10  20
2 4 3 0 5 6
8 10 20
8 10
4 5 10
4 5 10
Add 5+20=25
2 4 3 0 5 6
8 10 20
4 5 10 25
Add -6+50=44
2 4 3 0 5 6
2 4 3 0 5 6
8 10 20 50
8 10 20 50
4 5 10 25 44
4 5 10 25
The last row now represents the quotient and the remainder. The last entry of the bottom row (44 in this
case) is the remainder. The other numbers of the last row represent the coefficients of the quotient.
2 4 3 0 5 6
8 10 20 50
4 5 10 25 44
remainder r  44
coefficients of q( x)  4 x3  5x2  10 x  25
Note that synthetic division can only be used when the divisor, d ( x ) has the form  x  c  .
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Objective 3 Using the Remainder Theorem
The Remainder Theorem
If a polynomial f ( x) is divided by x  c , then the remainder is f (c ) .
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Objective 4 Using the Factor Theorem
By the Remainder Theorem if f (c)  0 , then the remainder when a polynomial f ( x) is divided by
x  c must be zero. Conversely, if x  c is a factor of f ( x) , then f (c ) must equal zero. The last two
statements can be rewritten as the following important theorem known as the Factor Theorem.
The Factor Theorem*
The polynomial x  c is a factor of the polynomial f ( x) if and only if f (c)  0 .
* Note that the phrase “if and only if” was used in the Factor Theorem. This phrase means that both the statement and its converse
are true. Therefore, if the polynomial x  c is a factor of the polynomial f ( x ) , then f (c)  0 , and if f (c)  0 , then the
polynomial x  c is a factor of the polynomial f ( x ) . The “if and only if” phrase is often written as “iff” in mathematics.
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Objective 5 Sketching the Graph of a Polynomial Function