1. 4-3 The Remainder and Factor Theorems
Chapter 4: Polynomial and Rational Functions
4-3 The Remainder and Factor Theorems

2. 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.

3. 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
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
Step 3
Bring down the first coefficient. -1 2

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

5. 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
Step 7 -2 2 -8 2 -2 8 1
2x2 – 2x + 8
Remainder 1
The quotient is

6. 4-3 The Remainder and Factor Theorems
HW 4-3 pg. 219 #15-39 odd

7. 4-3 The Remainder and Factor Theorems
Example 1: Divide by using synthetic division.
(x4 – 3x2 + 6)  (x + 1)

8. 4-3 The Remainder and Factor Theorems
Problem 1: Divide by using synthetic division.
(x3 + 2x2 – 5x – 6)  (x – 2)

9. 4-3 The Remainder and Factor Theorems
Problem 2: Divide by using synthetic division.
(x4 – x3 + 2x + 5)  (x + 2)

10. 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)

11. 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)