Download Lecture notes for Section 5.3

Int. Alg. Notes
Section 5.3
Page 1 of 5
Section 5.3: Dividing Polynomials; Synthetic Division
Big Idea: Polynomials are the most important topic in algebra because any equation that can be written using
addition, subtraction, multiplication, division, integer powers, or roots (which are rational powers) can be
solved by converting the equation into a polynomial equation. The third step toward acquiring this awesome
power is to be able to divide polynomials.
Big Skill: You should be able to divide polynomials using long division and, when appropriate, synthetic
division.
Dividing a polynomial by a monomial: Divide the monomial into each term of the polynomial, and cancel
when possible.
Practice:
24 z 5
1.

18 z 2
2.
9 p 4  12 p3  3 p 2

3p
3.
x 4 y 4  8 x 2 y 2  4 xy

4 x3 y
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Int. Alg. Notes
Section 5.3
Page 2 of 5
Dividing a polynomial by a polynomial using long division:
Long division of polynomials is a lot like long division of numbers:
a. Arrange divisor and dividend around the dividing symbol, and be sure to write them in
descending order of powers with all terms explicitly stated (even the terms with zero
coefficients).
b. Divide leading terms, then multiply and subtract.
c. Repeat until a remainder of order less than the divisor is obtained.
Compute 579 ÷ 16
Comparison between dividing integers and dividing polynomials
Dividend
Remainder
 Quotient 
Divisor
Divisor
Compute (5x2 + 7x +9) ÷ (x + 6)
Practice:
6 x3  7 x 2  6 x  6
1.

2x 1
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Int. Alg. Notes
2.
Section 5.3
Page 3 of 5
8  9 x  2 x 2  12 x3  5 x5

x2  3
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) !!!
Synthetic division is a shorthand way to divide a polynomial by the linear factor x – c:
a. Write c outside the division bar and the coefficients of the dividend inside the bar.
b. Bring the leading coefficient of the dividend straight down.
c. Compute c times the number in the bottom row, and write the answer in the middle row to the
right.
d. Add and repeat until all coefficients are used up.
Comparison between long and synthetic division of polynomials
Compute (2x – 3x – 4x + 11) ÷ (x – 2) using long
Compute (2x3 – 3x2 – 4x + 11) ÷ (x – 2) using
division.
synthetic division.
3
2
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Int. Alg. Notes
Section 5.3
Page 4 of 5
Practice:
2 x3  x 2  7 x  13
1.

x2
2.
x 4  8 x3  15 x 2  2 x  6

x3
3.
3x 2  4 x  7

2x  5
Definition: Quotient of Functions
If f and g are two functions, then the new function that can be made by taking their quotient is called
f  x
 f 
defined as:    x  
, provided g(x)  0.
g  x
g
f
, and is
g
Practice:
f
f
1. If f  x   x3  2 x2  4 x  5 and g  x   x  2 , find    x  and    3 .
g
g
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Int. Alg. Notes
Section 5.3
Page 5 of 5
The Remainder Theorem
If the polynomial P(x) is divided by x – c, then the remainder is the value P(c). This is because when we divide
a polynomial by x – c, the remainder must be of degree less than x – c, which means the remainder has degree
of zero, which is just a numeric constant R. So:
Dividend
Remainder
 Quotient 
Divisor
Divisor
P  x
R
 Q  x 
xc
xc
P  x
R 

 x  c   Q  x 
  x  c
xc
xc

P  x  Q  x  x  c  R
P c  Q c  c  c  R
R  P c
Practice:
1. Use the remainder theorem to find the remainder when f  x   3x3  2x  6 is divided by x  2 .
The Factor Theorem
If P(x) is a polynomial function, then x – c is a factor of P(x) if and only if R = P(c) = 0.
In other words, if P(c) = 0, then P(x) can be written as P(x) = (x – c)(Quotient(x)).
(This can be used to see if a divisor divides evenly into a dividend quickly.)
Practice:
1. Use the factor theorem to determine whether f  x   2x3  x2 16x  15 is divisible by x – 2 and x + 3.
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.