1. Remainder and
Factor Theorems

2. Some basics A polynomial of degree zero is a constant
A polynomial of degree 1 is called linear
e.g.
A polynomial of degree 2 is called quadratic
Degree 3 – cubic, 4 – quartic, 5 – quintic etc.
“Degree” is also sometimes called “order”
descending order
ascending order
unordered

3. Identities and Equations
An equation tells you that something is true in a particular situation, not always!
Now look at this equation:
Always true
Certainly! The LHS and RHS are identically equal
This is called an “identity”

4. Equating coefficients
This may be obvious, but it’s very useful!
Example:
Identical polynomials have identical coefficients.
Note: an identity is an equality relation.
See page 83 text book.

5. Long Division of Polynomials

6. Long Division of Polynomials
Divide 6x2 – 26x + 12 by x – 4.
The dividend is 6x2 – 26x + 12 and the divisor is x – 4.
We begin by arranging them as follows:

7. Long Division of Polynomials
The last line then contains the remainder.
The top line contains the quotient.

8. Long Division of Polynomials
The result of the division can be interpreted as.

9. Long Division of Polynomials

10. Long Division of Polynomials
We summarize the long division process in the following theorem.

11. Long Division of Polynomials

12. Long Division of Polynomials

13. Long Division of Polynomials

14. Long Division of Polynomials

15. Factor Theorem

16. Factor Theorem

17. Factor Theorem

18. Synthetic Division

19. Synthetic division is a quick method of dividing polynomials.
It can be used when the divisor is of the form x – c.
In synthetic division, we write only the essential parts of the long division.

20. Long Division vs. Synthetic Division
Compare the following long and synthetic divisions, in which we divide 2x3 – 7x by x – 3.

21. Long Division vs. Synthetic Division

22. Thus, 2x3 – 7x2 + 5 = (x – 3)(2x2 – x – 3) – 4.
Synthetic Division
From the last line, we see that the quotient is 2x2 – x – 3 and the remainder is –4.
Thus, 2x3 – 7x2 + 5 = (x – 3)(2x2 – x – 3) – 4.