Example 1

divisor

dividend

quotient

remainder

Remainder Theorem: The remainder is the value of the function evaluated for a given value.

Example 2 – Synthetic Division

(All exponents must be accounted for; zero is inserted in for x1.)

1

1. Drop the first coefficient down.2. Multiply by the divisor &

place under next coefficient, then add.

- 1

- 2 3. Repeat with all coefficients.

2

2

- 2

0

4. Place a vertical bar before the last value, separating the quotient from the remainder.5. The values to the left of the bar are the

coefficients of a polynomial of a degree one less than the original polynomial.

Factor Theorem: If the remainder of this process is zero, the quotient is a factor of the original polynomial. Meaning it is one polynomial that was multiplied with others resulting in the original polynomial.

(Another name for the quotient is a “depressed polynomial”)

Example 3

Long division or synthetic division could also be used. If the remainder is zero, then x – 1 is a factor.

Since f(1) = 0, x – 1 is a factor of the polynomial.

Example 4(without graphing!)

1. List all the factors of the constant:1 6

2 3

-1 -6

-2 -32. Evaluate each factor in the

polynomial:

Because the polynomial is a degree of three, there should be three binomial factors. These binomial factors consist of the constant factors that resulted in a value of zero:

1, 2, -3, so the binomial factors are:

(x – 1), (x – 2), & (x – 3) You can verify the results by multiplying it out.

Example 5

HW: Page 226