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ALGEBRA II
LESSON 7-4
REMAINDER AND FACTOR
THEOREMS
Pg. 365-370
Today’s Key Concepts
- Be able to EVALUATE the division of a polynomial by a binomial
- Determine whether a binomial is a factor of a polynomial
TERMS: 1. Synthetic Division: A method used to divide a polynomial by a binomial
2. Synthetic Substitution: Using Synthetic Division to evaluate a function
3. Long Division: Method of Dividing large numbers used in grade school
Long Division Example:
_ 214_
15 3210
30
21
15
60
60
Polynomial Long Division:
x2 + 3x + 2
x – 1x3 + 2x2 – x – 2
x3 - x2
3x2 – x
3x2 - 3x
2x –2
2x -2
Synthetic Division:
(Same Problem)
+1  1 + 2 – 1 – 2
__ 1_+3 +2_
1 3 2 0
x2 + 3x + 2
What is Synthetic Division? Like Long Division of grade school revised for polynomials!
RULES FOR SYNTHETIC DIVISION
1) Always divide by the zero of the given binomial factor (or just remember to switch the sign)
(Example: If dividing by x – 2 , divide by +2)
2) Always use PLACEHOLDERS for missing coefficients of a descending order polynomial
(Example: If x4 – 3x2 + 4x + 6 is given, use 1+ 0 – 3 + 4 + 6 on the first line
3) Remove the variables for this style of division on the first line
4) The Leading Coefficient drops directly to the solution line to start the process
5) Multiply the divisor by the Leading Coefficient and add to the second line under the second number
6) Repeat Step 5 until all numbers are accounted for
7) The solution line displays a list of numbers that start with an attached variable of one less degree than the
original (this is called Depressed Polynomial)
**Note: If the last number is a zero, the Divisor’s Binomial is a factor of that polynomial! (Known as
the Factor Theorem)
Factor Theorem: “With a polynomial function, IF the binomial (x –n) is a factor of that polynomial f(x),
THEN f(n) = 0”