4-3 The Remainder and Factor Theorems Chapter 4: Polynomial and Rational Functions 4-3 The Remainder and Factor Theorems

4-3 The Remainder and Factor Theorems Remainder Theorem: If a polynomial P(x) is divided by x-r, the remainder is a constant, P(r), and P(x) = (x-r)  Q(x) + P(r) where Q(x) is a polynomial with degree on less than the degree of P(x) P(x)=2x3+6x+9 (x−r)=x+1 Q(x)=2x2−2x+8 P(r)=1 Factor Theorem: The binomial x-r is a factor of the polynomial P(x) if and only if P(r) = 0.

4-3 The Remainder and Factor Theorems Synthetic division is a shortcut for dividing a polynomial by a binomial of the form x-r. The step for dividing 2x3 + 6x + 9 by x + 1 using synthetic division are shown below. 2x3 + 6x + 9 2x3 + 0x2 + 6x + 9 2 0 6 9 Arrange the terms of the polynomial in descending powers of x. Insert zeros for any missing powers of x. Write the coefficients as shown. Step 1 Write the constant r of the divisor x-r. In this case, write -1. Step 2 -1 2 0 6 9 Step 3 Bring down the first coefficient. -1 2 0 6 9 2

4-3 The Remainder and Factor Theorems Multiply the first coefficient by r. Then write the product under the next coefficient. Add. -1 2 0 6 9 Step 4 -2 2 -2 Multiple the sum by r. Then write the product under the next coefficient. Add Step 5 -1 2 0 6 9 -2 2 2 -2 8 Step 6 Repeat Step 5 for all coefficients in the dividend. -1 2 0 6 9 -2 2 -8 2 -2 8 1

4-3 The Remainder and Factor Theorems The final sum represents the remainder, which in this case is 1. The other numbers are the coefficients of the quotient polynomial, which has a degree one less than the dividend. -1 2 0 6 9 Step 7 -2 2 -8 2 -2 8 1 2x2 – 2x + 8 Remainder 1 The quotient is

4-3 The Remainder and Factor Theorems HW 4-3 pg. 219 #15-39 odd

4-3 The Remainder and Factor Theorems Example 1: Divide by using synthetic division. (x4 – 3x2 + 6)  (x + 1)

4-3 The Remainder and Factor Theorems Problem 1: Divide by using synthetic division. (x3 + 2x2 – 5x – 6)  (x – 2)

4-3 The Remainder and Factor Theorems Problem 2: Divide by using synthetic division. (x4 – x3 + 2x + 5)  (x + 2)

4-3 The Remainder and Factor Theorems Example 2: Use the Remainder Theorem to find the remainder for each division. State whether the binomial is a factor of the polynomial. (x3 – 2x2 – 33x + 10)  (x + 5)

4-3 The Remainder and Factor Theorems Problem 3: Use the Remainder Theorem to find the remainder for each division. State whether the binomial is a factor of the polynomial. (x3 – 7x + 5)  (x + 3)