When x² - 3x + 2k is divided by x + 2, the remainder is 7. Find the value of k. Use synthetic division to do so.

Well, synthetic division sounds serious, but let's see if we can bring some humor into it.

Now, let's keep things simple. We have this long polynomial x² - 3x + 2k, and we want to divide it by the "cool dude" x + 2.

First, let's rewrite the equation using the "long division dance" called synthetic division:

-2 | 1 -3 2k
|___________

We start by bringing down the first term, which is 1.

-2 | 1 -3 2k
|___________
1

Now, we have to multiply -2 by 1, which gives us -2. We write that below the next term and add them up.

-2 | 1 -3 2k
|___________
1
+ 2
____

Let's add -3 and 2, and we get -1. Hmmm, it's not 7 yet. We're not done, but we're getting there!

-2 | 1 -3 2k
|___________
1
+ 2
____
-1

Finally, we multiply -2 by -1, and we land with 2. We add it to 2k, but we know that should be equal to 7:

-2 | 1 -3 2k
|___________
1
+ 2
____
-1
+ 2
____
2k + 1

Now, we have 2k + 1 = 7. Time for a little math magic:

2k = 7 - 1
2k = 6

Divide both sides by 2:

k = 3

Ta-da! By using our clownish synthetic division skills, we find that k is equal to 3. I hope that brought a smile to your face!

To find the value of k, we can use synthetic division. Synthetic division allows us to divide a polynomial by a linear factor and determine the remainder.

The polynomial is x² - 3x + 2k, and we want to divide it by x + 2.

Step 1: Set up the synthetic division table:
-2 | 1 -3 2k
-----------------
? ?

Step 2: Bring down the coefficient of the highest power term, which is 1, and write it in the top row of the table:
-2 | 1 -3 2k
-----------------
1

Step 3: Multiply the divisor (-2) by the number in the bottom row (1), and write the result in the second row of the table, below the next coefficient of the polynomial:
-2 | 1 -3 2k
-----------------
1
-2

Step 4: Add the numbers in the second row to the corresponding numbers in the first row, and write the result in the third row of the table:
-2 | 1 -3 2k
-----------------
1
-2
-5

Step 5: Multiply the divisor (-2) by the number in the third row (-5), and write the result in the fourth row of the table, below the next coefficient of the polynomial:
-2 | 1 -3 2k
-----------------
1
-2
-5
10

Step 6: Add the numbers in the fourth row to the corresponding numbers in the third row, and write the result in the fifth row of the table:
-2 | 1 -3 2k
-----------------
1
-2
-5
10
5

Step 7: The number in the fifth row represents the remainder, which is given to be 7. Therefore, we have:
5 = 7

Step 8: Solve the equation 5 = 7:
5 = 7
This equation is not true for any value of k.

Therefore, there is no value of k that satisfies the given condition.

To find the value of k, we can use synthetic division.

Step 1: Write the given polynomial in the form of (x + 2) without the remainder:
x² - 3x + 2k = (x + 2)(x - a) + 7

Step 2: Set up the synthetic division table with -2 (opposite sign of the divisor x + 2) at the top and write the coefficients of the polynomial underneath.

-2 | 1 -3 2k
_______________________

Step 3: Bring down the first coefficient, which is 1.

-2 | 1 -3 2k
1

Step 4: Multiply the divisor (-2) by the value you brought down (1) and write the result below the next coefficient.

-2 | 1 -3 2k
1
-2

Step 5: Add the result to the next coefficient.

-2 | 1 -3 2k
1
-2
____
-5

Step 6: Repeat steps 4 and 5 until you reach the last coefficient.

-2 | 1 -3 2k
1
-2
____
-5 12k

Step 7: The last number in the synthetic division table represents the remainder, which is given as 7 in the question.

Therefore, we have -5 + 12k = 7.

Step 8: Solve for k.

-5 + 12k = 7
12k = 12
k = 1

Hence, the value of k is 1.

As you know from the Remainder Theorem, dividing p(x) by (x-a) gives f(a) as the remainder.

we have f(-2) = 7

f(-2) = 4 + 6 + 2k = 7
2k = -3
k = -3/2

So, we have x^2 - 3k - 3