1. Dividing Polynomials A review of long division:
Long division of polynomials
Ex 1: Divide the polynomials

2. Division Algorithm
If P(x) and D(x) are polynomials, with D(x)  0, then there exist unique polynomials Q(x) and R(x); where R(x) is either 0 or of degree less than the degree of D(x), such that
The polynomial P(x) is called the dividend.
The polynomial D(x) is called the divisor.
The polynomial Q(x) is called the quotient.
The polynomial R(x) is called the remainder.

3. Notice in long division, we always use
Notice in long division, we always use multiplication to make the first term of the divisor equal the dividend and then subtract this product from the dividend.
Ex 2: Divide the polynomials using long division

4. Synthetic Division
In synthetic division, we use only the coefficients of the dividend and the opposite of the number of the linear expression that we are dividing by.
Ex 3: Divide the polynomials using synthetic division

5. Remainder Theorem
If the polynomial P(x) is divided by x - c, then the remainder is the value of P(c).
Ex 4: Use the remainder theorem to find P(-2) for

6. Factor Theorem
c is a zero of P if and only if x - c is a factor of P(x).
Ex 5: Let Show that
P(1) = 0, and use this fact to factor P(x) completely.
Ex 6: Find a polynomial of degree 4 that has zeros -3, 0, 1, and 5.

7. Assignment S 4.2: pg
#6,8,9,16,18,20,32-34,47-49,55,56