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

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.

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

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

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

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.

Assignment S 4.2: pg 331-332 #6,8,9,16,18,20,32-34,47-49,55,56