1. n n – 1 f (x) = an x n + an – 1 x n – 1 +· · ·+ a 1 x + a 0 a 0 a0
OFFICIAL POLYNOMIAL JARGON
A polynomial function is a function of the form
f (x) = an x n + an – 1 x n – 1 +· · ·+ a 1 x + a 0
a 0 a0
constant term
an  0 an
leading coefficient
descending order of exponents from left to right. n
n – 1 n
degree
Where an  0 and the exponents are all whole numbers.
For this polynomial function, an is the leading coefficient,
a 0 is the constant term, and n is the degree.
A polynomial function is in standard form if its terms are
written in descending order of exponents from left to right.

2. 3 7 8 4 0 Let’s remember how to do long division. Remainder = 12 6 1 3 3 6 1 8 4
-1 8
Solution is:
- 3 1
Remainder = 1
- 9 1

3. x x2 + 2x - 6 (x + 4) x3 + 6x2 + 2x + 10 + 2x + 10 x3 + 4x2 2x2 2x2
So now, we’re going to use the same process to divide polynomials.
determine what to multiply by.
Use the 1st term to x x2
+ 2x
- 6
(x + 4)
x3 + 6x2 + 2x + 10
+ 2x
+ 10 x3
+ 4x2
2x2
x times what equals x3?
2x2
+ 8x
- 6x
x times what equals 2x2?
- 6x
- 24 34
x times what equals –6x?
This is the remainder!
Solution is: x2 + 2x – 6 +

4. You try . . . 2x2 + 3x + 2 (x + 6) 2x3 + 15x2 + 20x + 10 2x3 + 12x2
x times what equals 2x3?
3x2
+ 18x 2x
x times what equals 3x2? 2x
+ 12 -2
x times what equals 2x?
This is the remainder!
Solution is: 2x2 + 3x

5. Using Synthetic Division
Another way to divide a polynomial is to use
synthetic division.
Use synthetic division to divide
2 x x x - 7 by (x – 3).
This is in the format (x – k).
What is k if you’re dividing by
(x + 7) ? -7
So, k = 3

6. Using Synthetic Division
SOLUTION
2 x x 3 + (–8 x 2) + 5 x + (–7)
Polynomial in
standard form
Polynomial in standard form
2 0 –8 5 –7 3
k-value
3 •
Coefficients
Coefficients 6 18 30
105 2 35 6 10 98
Solution!
X3-term
X2-term
Constant
Remainder
x-term
+ 2 x3
+ 6 x2
+ 10 x
+ 35

7. You Try
Divide 3x4 + 4x3 – 8x – by (x + 2)
3 x x x x
Polynomial in
standard form
Polynomial in standard form -2
k-value
-2 •
Coefficients
Coefficients -6 4 -8 32 3
-16 -2 4 12
Solution!
X3-term
X2-term
X-term
Constant
Remainder
- 16
+ 3 x3
- 2 x2
+ 4 x

8. --The divisor must be of the form (x – k).
A Few Notes
--The divisor must be of the form (x – k).
--If it is not, then you must use long division (i.e. 2x – 7).
--If the remainder is 0, then the answer is a factor of the
original polynomial!
Divide x2 + 7x – by (x – 1) 1 8
Remainder
x-term
constant
x + 8
(x – 1)(x + 8) = x2 + 7x – !!
--Something really cool  f(k) = the remainder !!

9. Try some more.
Find f (4) by using synthetic division f (x) = x x - 8
Divide x x x (x + 2)
Divide x x (2x - 1)

10. 1 1 1 -14 3 2 • 2 6 14 1 7 3 k-value Solution! Some Vocab
Divide x x x (x - 2)
Polynomial in
standard form
Polynomial in standard form 3
k-value
2 •
Coefficients 2 6 14 1 7 3
Solution!
Remainder of zero means
(x2 + 3x + 7) is a factor, but
is also known as a
“Reduced Polynomial”.
X2-term
Constant
Remainder
x-term
+ x2
+ 3 x
+ 7