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Question 94 points

Express f(x) in the form f(x) = (x - k)q(x) + r for the given value of k.

f(x) = 3x3 - x2 + 2x + 5; k = -1

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Question 104 points

Express f(x) in the form f(x) = (x - k)q(x) + r for the given value of k.

f(x) = x3 - x2 + 3; k = -2

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Question 114 points

Express f(x) in the form f(x) = (x - k)q(x) + r for the given value of k.

f(x) = -6x3 + 2x2 + 5x - 10; k = 2

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Question 124 points

Express f(x) in the form f(x) = (x - k)q(x) + r for the given value of k.

f(x) = -5x4 + x3 + 2x2 + 3x - 1; k = 1

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Question 134 points

Use the remainder theorem and synthetic division to find f(k).

k = -3; f(x) = 2x3- 6x2 - 3x + 12

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Question 144 points

Use the remainder theorem and synthetic division to find f(k).

k = -3; f(x) = -5x5 - 3x3 - 5x2 - 5

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Question 154 points

Use the remainder theorem and synthetic division to find f(k).

k = -2; f(x) = x6 + 2x5 + 2x4 - 2x3 + 3x2 + 4x + 7

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Question 164 points

Use the remainder theorem and synthetic division to find f(k).

k = -4 + 3i; f(x) = x2 - 5x + 3

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Question 174 points

Use synthetic division to decide whether the given number k is a zero of the given polynomial function.

-3; f(x) = 8x3 - 4x2 + x + 255

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Question 184 points

Use synthetic division to decide whether the given number k is a zero of the given polynomial function.

1; f(x) = -x4 + 3x2 - x - 7

Explanation / Answer

P.S. The first four questions are having some sort of label "preview" which is not part of the problem to be solved, makes it difficult to understand the problem. Kindly resubmit these again for solving.

Ans 94: Second option.

If k = -1, then (x - k) = (x - (-1)) = x + 1.

So, options first, second an last options only need be considered.

However, if we try to conclude on the constant that results in each of the options, we see that only the third option gives us the constant as 3. The rest give some other constants. So our answer is

If we see the last option we have f(x) = (x + 1)(3x2 - 4x + 2) + 7 which will give us 9 as constant.

So we cannot consider this to be the correct option.

The first option results in a constant 5, but it does not have any negative coefficient, as is required. thus this option also cannot be right. Our correct option is the second option f(x) = (x + 1)(3x2 - 4x + 6) - 1 = 3x3 - 4x2 + 6x + 3x2 - 4x + 6 - 1 = 3x3 - x2 + 2x + 6.

Ans 104: Third option.

If k = -2, then (x - k) = (x - (-2)) = x + 2.

So, options need to be considered, since all have a factor x+2.

Now if we check what are constants that we get in each of the options we get 3 as a constant only for the third option, as below:

f(x) = (x + 2)(x2 - 3x + 6) - 9 = x3 - 3x2 + 6x + 2x2 - 6x + 12 - 9 = x3 - x2 + 3.

Ans 114: Third option.

If k = 2, then (x - k) = (x - 2) = x - 2.

So, options second and third to be considered, since only they have a factor x-2.

Now if we check what are constants that we get in each of those considered options we get -10 as a constant only for the third option, as below:

f(x) = (x - 2)(-6x2 - 10x - 15) - 40 = -6x3 - 10x2 - 15x + 12x2 + 20x + 30 - 40 = -6x3 + 2x2 + 5x- 40.

Ans 124: First option.

If k = 1, then (x - k) = (x - 1) = x - 1.

So, all options need to be considered, since all have a factor x-1.

Now if we check what are constants that we get in each of those considered options we get -1 as a constant only for the first option, as below:

f(x) = (x - 1)(-5x3 - 4x2 - 2x + 1) = - 5x?4 - 4x3 - 2x2? + x + 5x3 + 4x2 + 2x - 1 = - 5x4 + x3 + 2x2 + 3x - 1

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