Remainder Theorem

1. Remainder Theorem If the polynomial f(x) is divided by x  c, then the remainder is equal to f(c). Example 1. In other words, the remainder after performing synthetic division is the same number we would get if we replaced c into the polynomial and evaluated the polynomial. • Find f(2) by evaluating f(2) directly. • Find f(2) using synthetic division and the • remainder theorem. 13 Using synthetic division, let c = -2. Your Turn Problem #1 Both methods give a result of 13. 5

2. Find f(2) by evaluating f(2) directly. • Find f(2) using synthetic division and the • remainder theorem. Using synthetic division, let c = 2. 2 Both methods give a result of 61. Your Turn Problem #2 4 8 16 32 64 32 61 4 8 16

3. Factor Theorem A polynomial f(x) has a factor x  c if and only if f(c) = 0. We can use two different methods to answer the question. Either evaluate f(c) directly or use synthetic division. We’ll first evaluate f(c) directly. In other words, if the remainder after performing synthetic division is zero or the result from evaluating the polynomial at c is zero, then x  c is a factor. Answer: Yes. If f(c) = 0, then the divisor is a factor of the polynomial Show by using synthetic division. We are given x + 3. Therefore use c = 3. 1 0 Answer: Yes, f(3) = 0. Next Slide We are given x  3. Therefore use c = 3. The other method we can use is synthetic division. Often this method is more preferable because we can obtain more information than just the remainder.

4. Your Turn Problem #3 Show by evaluating f(c) directly. Answer: No, f(-3) =-6. 1 0 Show using synthetic division. Answer: Yes, since f(2)=0.

5. Let’s look further at the last your turn problem. Recall from the previous section, the bottom row gives us the quotient. 1 0 We can then write the polynomial completely factored as: Procedure: To factor a polynomial P(x) given a factor x  c. • The polynomial P(x) completely factored = (x  c)(quotient factored) Next Slide • Use synthetic division to show x  c is a factor of P(x) by showing remainder = 0. 2. Rewrite the quotient in proper form with variables. • Factor the quotient (if possible) using previous factoring techniques.

6. Example 4. Show g(x) is a factor of f(x) and complete the factorization of f(x). Solution: 1 0 Your Turn Problem #4 Show g(x) is a factor of f(x) and complete the factorization of f(x). 1st, show g(x) is a factor of f(x). 2nd, rewrite the quotient in proper form. 3rd, factor the quotient. Show the complete factorization.

7. Example 5. Show g(x) is a factor of f(x) and complete the factorization of f(x). Solution: 6 0 Your Turn Problem #5 Show g(x) is a factor of f(x) and complete the factorization of f(x). 1st, show g(x) is a factor of f(x). 2nd, rewrite the quotient in proper form. 3rd, factor the quotient. Show the complete factorization.

8. Example 6. Show g(x) is a factor of f(x) and complete the factorization of f(x). Solution: 1 0 0 0 16 16 1 0 0 0 0 Your Turn Problem #6 Show g(x) is a factor of f(x) and complete the factorization of f(x). The End. B.R. 3-5-07