1. Aim: What are the remainder and factor theorem?
Do Now:
Divide p(x) = x3 – 7x – 6 by x – 4
then write the result as
P(x) = D(x)Q(x) + R(x)

3. This is called Remainder Theorem
then f(x) = x3 – 7x – 6 = (x – 4)(x2 + 4x + 9) + 30.
If we use 4 to replace f(x) we get
f(4)=43 – 7(4) – 6 = 64 – 28 – 6 = 30
When f(x) is divided by x – a, the remainder will be equal to f(a)
This is called Remainder Theorem

4. Using the Remainder Theorem, find the remainder when f (x) = 3x4 + 2x3 + 4x is divided by x + 5

5. When f(x) is divided by x – a, if the remainder is 0, that means f(x) is divisible by x – a. In other word, x – a is a factor of f(x).
We can verify if x – a is a factor of f(x) by finding the value of f(a)
If f(a) = 0, then x – a is a factor of f(x) or we can say f(x) is divisible by x – a
This is called Factor Theorem

6. Use the Factor Theorem to determine whether x – 2 is a factor of f (x) = 3x7 – x4 + 2x3 – 5x2 – 4.
The remainder is non-zero. Then x = 2 is not a zero of f (x).

7. Determine whether f(x)= x6 + 5x5 + 5x4 + 5x3 + 2x2 – 10x – 8 is divisible by x + 4
The remainder is zero. Then f(x) is divisible by x + 4

8. Use the Factor Theorem to determine whether x – 1 is a factor of f (x) = 2x4 + 3x2 – 5x + 7.
     
f(1) = 7
then x – 1 is not a factor of f (x).

9. x – 5 is not a factor of 5x4 + 16x3 – 15x2 + 8x + 16.
Using the Factor Theorem, verify if x – 5 is a factor of f (x) = 5x4 + 16x3 – 15x2 + 8x       
f(5) ≠ 0
The remainder is zero, so, by the Factor Theorem:
x – 5 is not a factor of 5x4 + 16x3 – 15x2 + 8x + 16.

10. f(x) = 2x4 +7x3 – 4x2 + kx – 18 If x – 2 is a factor of f(x), find k
If x – 2 is a factor of f(x), that means f(2) = 0
f(2) = 2(16) + 7(8) – 4(4) + 2k –18 = 0
–16 + 2k –18=0 2k = – 54, k = – 27

11. When f(x) = 3x3+ kx2 + 5x + 6 is divided by x + 2, the remainder is –12. Find k
If the remainder is –12, that means f(– 2) = –12
f(-2) = –24 + 4k – = –12 –28 + 4k = –12 4k =16, k = 4