6 7 The Division Algorithm The Remainder Theorem

6 -7 The Division Algorithm & The Remainder Theorem 21 = 4 5 + 1. remainder dividend=quotient. divisor + remainder If a polynomial f(x) is divided by x - c, the remainder is the constant r(x), and f(x) = q(x). (x - c) + r(x) where q(x) is a polynomial with degree one less than the degree of f(x). r(x) = f(c)

► Synthetic Substitution: When synthetic division is used to evaluate a function. ► If ► Find f(6) ► Method 1: Synthetic Substitution: use synthetic division 6 2 2 ► Method 0 -5 8 -7 12 72 402 2460 12 67 410 2453 2: Direct Substitution

Example Problem 1 Find the remainder of divided by x + 1 Use the Remainder Theorem!!!

Factors of Polynomials ► Depressed Polynomial: When you divide a polynomial by one of its binomial factors, the quotient is called a depressed polynomial. Depressed Polynomial

The Factor Theorem The binomial x - c is a factor of the polynomial f(x) if and only if f(c) = 0. Example Problem 2 Use the factor theorem to determine whether the function has the factor x + 3. Then find the remaining factors. See if f(-3) = 0, cause if it does then x + 3 is a factor

Using Synthetic Division, you can find the depressed polynomial and then factor from there. -3 1 Factor: x + 3 1 Factored down to… 6 -1 -30 -3 -9 30 3 -10 0

Homework Assignment #37 p. 359 11 -29 odd, 34 -37, 46 -47