Algebra 2 Notes – Dividing Polynomials
A. Long Division and Synthetic Division
Division of Polynomials
Given polynomials p  x  and d  x   0 , there exists unique polynomials q  x  and r  x  such that
p  x  d  x q  x  r  x
Where r  x   0 or r  x  is less than the degree of d  x  . d  x  is the divisor, q  x  is the
quotient, and r  x  is the remainder.
Dividing a polynomial by a polynomial using long division:
Comparison between dividing integers and dividing polynomials
Dividend
Remainder
 Quotient 
Divisor
Divisor
Compute 579 ÷ 16
Compute (5x2 + 7x +9) ÷ (x + 6)
1.
6 x3  7 x 2  6 x  6

2x 1
2.
8  9 x  2 x 2  12 x3  5 x5

x2  3
Algebra 2 Notes – Dividing Polynomials
Dividing a polynomial by a binomial using synthetic division: THIS IS A SHORTCUT
THAT ONLY WORKS WHEN THE DIVISOR IS A LINEAR BINOMIAL (I.E., THE
DIVISOR IS x – c) !!!
Comparison between long and synthetic division of polynomials
Compute (2x – 3x2 – 4x + 11) ÷ (x – 2) using
Compute (2x3 – 3x2 – 4x + 11) ÷ (x – 2) using
long division.
synthetic division.
3
2 x3  x 2  7 x  13
1.

x2
f  x 3. 2x3  x2 16x  15 ÷ (x – 2)
x 4  8 x3  15 x 2  2 x  6
2.

x3
f  x 4. 3x3  2x  6 ÷ ( x  2 )
Algebra 2 Notes – Dividing Polynomials
B. The Remainder Theorem
The Remainder Theorem
If a polynomial p  x  is divided by  x  c  using synthetic division, then the remainder is p(c).
Examples:
a)
b)
x 4  2 x3  5 x  4
x 1
2 x3  3x 2  10
x2
c) f(x) = x3 + 2x2 - 5x + 7
f(1) =
C. The Factor Theorem
The Factor Theorem
For a polynomial p  x  , p  c   0 if and only if  x  c  is a factor of p  x  .
Examples: Determine if the following binomials are factors of f(x).
a) ( x3 – 14x + 8) and (x + 4)
b) (x4 – 6x3 – 40x + 33) and (x – 7)