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7/13/2012
OPERATIONS ON POLYNOMIALS
OPERATIONS ON
POLYNOMIALS
TYPES of Polynomials
According to the number of terms
continuation…
•
•
•
•
MONOMIAL – one term
BINOMIAL – two terms
TRINOMIAL – three terms
MULTINOMIAL – more than three terms
OPERATIONS ON POLYNOMIALS
OPERATIONS ON POLYNOMIALS
TYPES of Polynomials
COEFFICIENTS
According to degrees
A term is composed of
• Numerical coefficient
• Literal coefficient
7
•CONSTANT – degree 0
x yz
•LINEAR – degree 1
xy  x
•QUADRATIC – degree 2
•CUBIC – degree 3
3  x  3 x 2  x3
OPERATIONS ON POLYNOMIALS
SIMILAR Terms
• are terms that have the same literal coefficient.
Example: Identify the numerical and literal
coefficients.
5 xy 2  7 x 2 y 3  4 x 2 y 2  2 x 2 y 2
OPERATIONS ON POLYNOMIALS
ADDITION of Polynomials
RULE: Identify similar terms; add their numerical
coefficients and affix common literal coefficients.
Example: Identify similar terms.
3xy 2  7 xy 3  12 x 2 y 2  x 2 y 2
Example: Simplify by adding similar terms.
5 x 2 y 2  2 xy 3  4 x 2 y 2  xy 3
 x 2 y 2  3xy 3
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OPERATIONS ON POLYNOMIALS
ADDITION of Polynomials
COLUMN Method
RULE: Arrange similar terms in column, then
add.
Example: Add the polynomials below.
5x  7 x  x  4
3
Example:
Add the following polynomials:
1. x 2 y  2 x 2  y 2  xy 3 ; x3 y  3x 2  x 2 y  y 2
Solution:
x 2 y  2 x 2  y 2  xy 3
2
x3 y  x 2 y  3x 2  y 2
3x  8 x  3x  7
3
2
3
x3 y
2
8 x  x 4 x 11
 xy 3
 x2
Answer: x3 y  x 2  xy 3
Example:
OPERATIONS ON POLYNOMIALS
SUBTRACTION of Polynomials
Add the following polynomials:
2. 3x 4  5 x 2  x  12;  5 x 4  x3  11x  3
Solution:
Example: Subtract the 2nd polynomial from the first
one.
3 x4
 5 x2  x  12
5 x4  x3
 11x  3
4
RULE: a – b = a + (-b)
4 x3  7 x 2  2 x  4
7
3x3 
 8x2 
 3x 
2
3
2 x  x 5 x 10 x 9
x3  15 x 2  x  3
Answer: - 2 x4  x3  5 x2  10 x  9
Example:
Subtract the
Example:
2nd
from the
1st:
Subtract the 2nd from the 1st:
1. x 2 y  2 x 2  y 2  xy 3 ; x3 y  3x 2  x 2 y  y 2
Solution:
x y  2 x  y  xy
2

2
2
2. 3x 4  5 x 2  x  12;  5 x 4  x3  11x  3
Solution:
3
x3 y  x 2 y  3x 2  y 2
 x3 y 2 x 2 y 5 x 2 2 y 2  xy 3
Answer: - x3 y  2 x2 y  5 x2  2 y 2  xy3

3 x4
 5 x2  x  12
5 x4  x3
 11x  3
8 x4  x3 5 x 2 12 x 15
Answer: 8x4  x3  5 x2  12 x  15
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OPERATIONS ON POLYNOMIALS
MULTIPLICATION of Polynomials
Multiply:
RULE: a(b + c) = ab + ac
1. x3  3x 2  x  1; x  2
Solution:
Example: Multiply the polynomials below.
8 x
2
 7  4 x  7 x  2 x  4 
3
Example:
2


x 4 3 x3  x 2  x
2 x36 x 2 2 x 2
3 4
2
8 xx2 5 4x56
x 7x16
x23 x 
 32
324x 2


3
2
3 x  14
 28
57x 4x49

7 x 2 3 x2x28
4
4
 32 x  56 x  12 x  17 x 2  14 x  28
x3  3 x 2  x  1
x2
Answer: x4  5 x3  7 x2  3x  2
OPERATIONS ON POLYNOMIALS
DIVISION of Polynomials
x 6 
Long Division
Synthetic Division
Example: Perform the indicated operation.
x
2
OPERATIONS ON POLYNOMIALS
LONG DIVISION

 5 x  6   x  1
12
x1
x  1 x2  5 x  6
 x2  x
6 x 6
   6 x 6
12
OPERATIONS ON POLYNOMIALS
SYNTHETIC DIVISION
polynomial
Used when we have:
xa
1.
2
x  5x  6
Example:
x 1
SYNTHETIC DIVISION
Write the coefficients
ONLY of the variable
(arrange in decreasing
power/exponent)
5    6 2. Write a at the left side:
3. Bring down the
6 leading coefficient:
1
1 6 12
x  6 r.12  x  6  12
1 1
OPERATIONS ON POLYNOMIALS

Example: Perform the indicated division using
synthetic division.
x3  3x 2  x  3
x2
4
x  3x  x 2  4
2)
x2
1)
x 1
3
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OPERATIONS ON POLYNOMIALS
OPERATIONS ON POLYNOMIALS
SYNTHETIC DIVISION
SYNTHETIC DIVISION
x  3x  x  3
1.
x2
2.
3
2
2
3  1 3

2  2  6
1 1 3 3
x 2  x  3 r.  3
 x 2  x  3  x32
1

MORE Examples
OPERATIONS ON POLYNOMIALS
Find: a.) sum of the expressions
b.) subtract the second from the first.
1.2 x  y  5 and 3 y  2 z  4
a. 2 x  y  5    3 y  2 z  4 
 2x  4 y  2z 1
MORE Examples
OPERATIONS ON POLYNOMIALS
Find: a.) sum of the expressions
b.) subtract the second from the first.
2.   x  2 y  z  and 3  x  y  2 z 
a.   x  2 y  z   3  x  y  2 z 
  x  2 y  z  3x  3 y  6 z
 2x  5 y  7z
x 4  3x  x 2  4
x2
2
0 1 3 4
 
18
2   4  6
3 9 22
1 2
3
2
x  2 x  3 x  9 r. 22
 x3  2 x 2  3 x  9 
1

MORE Examples
22
x2
OPERATIONS ON POLYNOMIALS
Find: a.) sum of the expressions
b.) subtract the second from the first.
1.2 x  y  5 and 3 y  2 z  4
b. 2 x  y  5    3 y  2 z  4 
 2x  2 y  2z  9
MORE Examples
OPERATIONS ON POLYNOMIALS
Find: a.) sum of the expressions
b.) subtract the second from the first.
2.   x  2 y  z  and 3  x  y  2 z 
b.   x  2 y  z   3 x  y  2 z 
  x  2 y  z  3x  3 y  6 z
 4 x  y  5 z
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OPERATIONS ON POLYNOMIALS
MORE Examples
4. Subtract the second from the first.
3. ADD the two polynomials.

8 y 4  5 y3  2 y 2
6
14 y 4  5 y 3  4 y 2  3 y  5
OPERATIONS ON POLYNOMIALS
MORE Examples
5. Remove all grouping symbols and combine
like terms


8 y
4
2 y 4  5 y 3
OPERATIONS ON POLYNOMIALS
7. Multiply x  xy  y and x  y .
2
OPERATIONS ON POLYNOMIALS
MORE Examples
6. Remove all grouping symbols and combine
 9 x  2 y  3x  y  x  2 y  4 x
 7x  y
OPERATIONS ON POLYNOMIALS
MORE Examples
8. Multiply a
2n
x 2  xy  y 2
 7a n  10 and a  1 .
a 2 n  7 a n  10
x y
a 1
x  x y  xy
x 2 y  xy 2  y 3
2
 3y  7
 9 x  2 y  3x   y  x   2 y  4 x  4 y 
 2 x  3 y  11x  7 y  2 x  11x  10 y
 13x  10 y
3
 6
 5 y3  2 y 2
9 x   2 y  3x    y   2 y  x    2 y  4  x  y  
 2 x  3 y  5 x  7 y  6 x 
2

like terms
2 x  3 y  5 x   7 y  6 x  
MORE Examples
 2 y2  3y 1
6 y4
 2 y2  3y 1
6 y4
OPERATIONS ON POLYNOMIALS
MORE Examples
2
a
 x3  y 3
a
2n
2 n 1
7a
n 1
10a
7a n 10
a 2 n1  a 2 n  7a n1  7a n  10a  10
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MORE Examples
OPERATIONS ON POLYNOMIALS
OPERATIONS ON POLYNOMIALS
SUMMARY
9. Divide  15 x 2 y 2  20 x3 y 3 by  5 x 2 y 2
3  4xy
10. Divide 2 x3  7 x 2  11x  4 by 2 x  1
x 2  3x  4
In this section you learned to perform the
following operations on polynomials:
•ADDITION
•SUBTRACTION
•MULTIPLICATION
•DIVISION
OPERATIONS ON POLYNOMIALS
End
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