Problem 1)
Write x^3-4 as 1x^3 + 0x^2 + 0x + (-4)x^0
The coefficients here are 1, 0, 0, and -4
note how the exponents step down one at a time (3, 2, 1, 0)
Write the coefficients 1, 0, 0, -4 in that order along a single row. Write the test root x = -2 in a box off to the left of the row of coefficients. See Figure1A (attached) to see how the set up is done so far.
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Now we will follow the steps to perform synthetic division. The first step is to drop the first coefficient (which is 1) down below the horizontal line. This is shown by the letter A in red. See Figure1B (attached).
We then multiply that dropped value (1) by the test root (-2) to get 1 times -2 = -2. This result of -2 is placed where you see the red letter B (Figure1B)
Now add straight down: 0 plus -2 = -2
The result -2 is placed just below the previous result. It is placed where you see the red letter C
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Repeat these steps over and over until the bottom row is filled out. By that, I mean we have the bottom row terms line up with the right most value in the top row. See figure1C to have a look at what I mean. We stop once we get to -12 in the bottom row as there are no more terms to fill out.
The last term in the bottom row (-12) is the remainder. The other terms help construct the quotient.
In this case, the quotient is 1x^2 + (-2)x + 4x^0 which simplifies to x^2 - 2x + 4
The remainder is -12
Put this all together and we have the final answer of [tex]x^2 - 2x + 4 - \frac{12}{x+2}[/tex]
So the final answer is choice D
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Problem 2)
Follow the same kind of steps shown in problem 1. This time we'll have the synthetic division table you see in the image Figure2 (attached)
The last value in the bottom row is the remainder. The remainder of 0 indicates (x+2) is a factor of 2x^3+x^2-2x+8
The quotient is the other values in the bottom row. The other values in that bottom row are (in order) 2, -3, 4 so the quotient is 2x^2 - 3x + 4
This makes the final answer to be choice C
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Problem 3)
We can use the remainder theorem and skip synthetic division
2x-1 = 0 leads to x = 1/2 = 0.5
Plug x = 0.5 into the function f(x) = 6x^2 + 11x - 3 to get
f(x) = 6x^2 + 11x - 3
f(0.5) = 6(0.5)^2 + 11(0.5) - 3
f(0.5) = 4
So the remainder is 4
Answer: 4
Side Note: if you wish to use synthetic division, then check out "Figure3" in the attachments
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Problem 4)
The answer is choice with the bottom row of 3, -13, and -54. This looks to be choice C. The same steps are followed as done in problems 1 through 3.
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Problem 5)
Plug x = -2 into f(x) = 6x^2 + 11x - 3
f(x) = 6x^2 + 11x - 3
f(-2) = 6(-2)^2 + 11(-2) - 3
f(-2) = -1
Final Answer: -1
