1) divide using synthetic division, and write a summary statement in fraction form.
two x to the fifth minus x to the fourth plus three x squared minus x plus five divided by quantity x minus one

a. 2x4 + x3 + 4x2 + 3x + eight divided by quantity x minus one
b. 2x4 + x3 - x2 + 2x + 1 + six divided by quantity x minus one
c. 2x4 + x3 + x2 + 4x + 3 + eight divided by quantity x minus one
d. 2x4 - 3x3 + x + six divided by quantity x minus one

2) divide f(x) by d(x), and write a summary statement in the form indicated.
f(x) = x4 - 4x3 + 2x2 - 4x + 1; d(x) = x2 + 1

a. f(x) = (x2 + 1)( x2 + 4x + 1)
b. f(x) = (x2 + 1)( x2 - 4x + 1) + 12x + 3
c. f(x) = (x2 + 1)( x2 + 4x + 1) + 12x + 3
d. f(x) = (x2 + 1)( x2 - 4x + 1)

3) find the remainder when f(x) is divided by (x - k).
f(x) = 3x3 - 4x2 - 3x + 14; k= 3

a. 50
b. 68
c. -12
d. 112

4) use synthetic division to determine whether the number k is an upper or lower bound (as specified) for the real zeros of the function f.
k = -1; f(x) = 4x3 - 2x2 + 2x + 4; lower bound?

a. yes
b. no

5) use the rational zeros theorem to write a list of all possible rational zeros of the function.
f(x) = -2x4 + 4x3 + 3x2 + 18

Question 1:

Refer to the first picture below
The synthetic division works by writing the coefficient only of the polynomials in question in order of the degree of the polynomials. 
Our polynomial expression is given: 2x⁵ - x⁴ + 3x² - x + 5
The coefficient in order are 2, -1, 3, 0, -1, 5
The value '0' in the list is for the polynomial x²

Then divide these coefficients by 1 ⇒ you get this value by x - 1 = 0, x = 1
The rest of the step is explained in the picture.
We have an answer  ⇒ The last term in fraction form is the remainder

Option C

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Question 2

We can use the usual way of Algebraic division as shown in picture 2 below
We have no remainder, so the answer to the question is given in the form of factors ⇒ (x² + 1)(x² - 4x + 1)

Option D
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Question 3

We are given the remainder to be k = 3 when we divide f(x) by (x - k)
So, f(3) = 3(3)³ - 4(3)² - 3(3) + 14 = 50

Option A

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Question 4

k is upper bound when f(x) is divided by (x - k) where k > 0 using the synthetic division and the answer gives all positive numbers

k is lower bound when f(x) is divided by (x - k) where k < 0 using the synthetic division and the answer gives an alternating sign

Refer to picture 3 below, the values of the answer have alternating signs (+) then (-) then (+) then (-) again. This means k = -1 is a lower bound

YES
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Question 5

Focus on the coefficient of the leading term which is -2x⁴ and the last term, 18.

Factors of -2 is 1, -1, 2, -2 ⇒ Let's call this group as 'p'
Factors of 18 is 1, -1, 2, -2, 3, -3, 6, -6, 9, -9, 18, -18 ⇒ This is group 'q'

Possible values for p ÷ q are
1, -1, 0.5, -0.5, 2, -2, 1.5, -1.5, 6, -6, 3, -3, 9, -9, 4.5, -4.5, 18, -18 ⇒ These are the list for possible roots.