(4x^3-9x+8x^2-18)/(x+2)-use synthetic division to divide. I set it up like this (pretend the parentheses are the synthetic division box).
(-2)
4 8 -27
4 0 -27.
For remainders, according to my book, we're supposed to divide the remainder by the divisor. So it would be -27/-2. Yet my book says there is no remainder. It says the answer is 4x^2-9. How did it get this?
Well Jeremiah, it's as simple as your forgetting the -9x term
try
(-2)
4 8 -9 -18
I get 4 0 -9 0
To perform synthetic division, you set up the dividend, divisor, and synthetic division box, as you correctly did:
Dividend: 4x^3 - 9x + 8x^2 - 18
Divisor: x + 2
Synthetic Division Box:
-2 | 4 8 -9 0
|_________
Next, you bring down the leading coefficient, which is 4, and multiply it by the divisor, which is -2:
-2 | 4 8 -9 0
|_______
|
| -8
Then, add -8 to the second term:
-2 | 4 8 -9 0
|_______
| -8
|
| -8
Next, multiply -8 by the divisor -2:
-2 | 4 8 -9 0
|_______
| -8
| 16
| -8
Add 16 to the third term:
-2 | 4 8 -9 0
|_______
| -8
| 16
| -8
Now, multiply -8 by the divisor -2:
-2 | 4 8 -9 0
|_______
| -8
| 16
| -16
| -8
Add -16 to the fourth term:
-2 | 4 8 -9 0
|_______
| -8
| 16
| -16
| 0
Finally, divide the resulting coefficients by the divisor -2:
-2 | 4 8 -9 0
|_______
| -8
| 16
| -16
| 0
This gives you the quotient: 4x^2 - 8.
So the long division result is:
4x^2 - 8
Since there is no remainder, we can ignore it.
Therefore, the correct answer is 4x^2 - 8, not 4x^2 - 9. It seems like there might be an error in your book.
To divide the polynomial (4x^3-9x+8x^2-18) by (x+2) using synthetic division, you set up the synthetic division box like this:
-2 | 4 8 0 -18
To begin, bring down the first coefficient, which is 4, to the bottom row:
-2 | 4
----
Next, multiply the divisor, -2, by the number you just brought down and write the result under the next coefficient:
-2 | 4
-8
----
Add the two numbers in the second row to get the new result:
-2 | 4 0
-8
----
-8
Repeat the process by multiplying the divisor, -2, by the new result (-8 in this case) and write the result under the next coefficient:
-2 | 4 0 -8
-8
----
-8
Add the numbers in the second row to get the new result:
-2 | 4 0 -8
-8
----
-8
Since we have reached the last coefficient, the numbers in the bottom row represent the coefficients of the quotient:
-2 | 4 0 -8
-8
----
-8
Therefore, the quotient is 4x^2 + 0x - 8, which simplifies to 4x^2 - 8.
There is no remainder in this case, which means that the polynomial is evenly divisible by (x + 2). The book's answer of 4x^2 - 9 appears to be incorrect because it does not align with the synthetic division results.