1. Rational
Zeros
Theorem
Upper &
Lower
Bounds
Long
Division
Remainder Theorem
Factor
Theorem
Synthetic
Division
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2. Division involves a dividend divided by a divisor to obtain a quotient and a _________.
remainder

3. 2x4-x3-x=(2x2+x+1)(x2-2)+(x-2)
What is the answer of 2x4-x3-2 divided by 2x2+x+1? Write in polynomial form.
2x4-x3-x=(2x2+x+1)(x2-2)+(x-2)

4. Divide f(x)=x2-2x+3 b x-1. Write in polynomial form.

5. Divide f(x)=x3+4x2+7x-9 by x+3 and write a summary statement in polynomial form.

6. f(x)=(x2-4x+12)(x2+2x-1)-32x+18
Divide f(x)=x4-2x3+3x2-4x+6 by x2+2x-, write a summary statement in fraction form.
f(x)=(x2-4x+12)(x2+2x-1)-32x+18

7. If a polynomial is divided by ___, then the remainder is r=f(k).
x-k

8. r=f(-1)=3(-1)2+7(-1)-20=3-7-20=-24
Using the remainder theorem, find the remainder when f(x)=3x2+7x-20 is divided by x+1.
r=f(-1)=3(-1)2+7(-1)-20=3-7-20=-24

9. What is the remainder when f(x)=2x2-3x+1 is divided by x-2?

10. Use the remainder theorem to find the remainder when you divide f(x)=x3-x2+2x-1 by x+3
-43

11. Use the remainder theorem to find the remainder when f(x)=2x3-3x2+4x-7 by x-25

12. A polynomial function f(x) has a factor x-k if and only if f(k)=_.

13. Using the factor theorem, factor f(x)=3x2+7x-20 by dividing it by the known factor x+4.

14. Is x-1 a factor of x3-x2+x-1?
Yes

15. Use the factor theorem to determine whether x-2 is a factor of x3+3x-4No

16. Use the factor theorem to determine where x+2 is a factor of 4x3+9x2-3x-10
Yes

17. This shortcut method for the division of a polynomial by a ______ _______ x-k is synthetic division.
linear divisor

18. Divide 2x3-3x2-5x-12 by x-3 using synthetic division and write a summary statement in fraction form.
2x3-3x2-5x-12 = 2x2+3x+4
x-3

19. Divide using synthetic division and write a summary statement in fraction form: x3-5x2+3x-2 x+1

20. Divide using synthetic division and write a summary statement in fraction form: 9x3+7x2-3x x-10

21. Divide using synthetic division and write a summary statement in fraction form: 5x4-3x+1 4-x

22. Real zeros of a polynomial function are either ________ zeros – zeroes that are rational numbers – or __________ zeros – zeros that are irrational numbers.
rational, irrational

23. Find the rational zeros of f(x)=x3-3x2+1
So f has no rational zeros.

24. (all positive & negative)
Use the rational zeroes theorem to write a list of all possible rational zeroes of f(x)=6x3-5x-1 1
1, 2, 3, 6
(all positive & negative)

25. (all positive & negative)
Use the rational zeroes theorem to write a list of possible rational zeroes: f(x)=2x3-x2-9x+9
1, 3, 9
1, 2
(all positive & negative)

26. Find all the zeroes of the function and identify each zero as rational or irrational: f(x)=2x4-7x3-2x2-7x-4
-1/2 and 4; rational
No irrational

27. We narrow our search for real zeros by using a test that identifies upper and lower ______ for real zeros.
bounds

28. Find a lower bound of f(x)=2x4-7x3-8x2+14x+8-2

29. Use synthetic division to prove that to number k is an upper bound for the real zeroes of the function: k=3; f(x)=2x3-4x2+x+3
Yes

30. Use synthetic division to prove that the number k is an upper bound for the real zeroes of the function f: k=2; f(x)=x4-x3+x2+x-12
Yes

31. Use synthetic division to prove that the number k is a lower bound for the real zeroes of the function f: k=-1; f(x)=3x2-4x2+x-2
Yes