I'm trying to finish up my summer math packet, and I'm almost done, but I'm having trouble figuring out this one:

"Divide and simplify these expressions."

(3x^4 - 2x^3 + 16x - 192)
/
(x^2 - 8)

so, that's 3x to the 4th, minus 2x cubed, plus 16x, minus 192 ---> all divided by x squared, minus 8

I'm checking w/ my textbook and other algebra sites to brush up on the principles, but I'm still having trouble with this one, so I'd appreciate any thoughts -thanks

you are doing "long algebraic division"

make sure you left a space for the missing x^2 term or insert 0x^2 in its place

I got it to divide exactly, and my answer is
3x^2 - 2x + 24

It is hard to show the algorith in this forum, but here is a nice webpage to show you how to do it

http://www.sosmath.com/algebra/factor/fac01/fac01.html

To divide and simplify this expression, you can use polynomial long division. Here's how you can do it step by step:

Step 1: Make sure the terms of both the dividend (3x^4 - 2x^3 + 16x - 192) and the divisor (x^2 - 8) are arranged in descending order of their exponents.

Step 2: Divide the first term of the dividend (3x^4) by the first term of the divisor (x^2) to get the first term of the quotient. In this case, (3x^4 / x^2 = 3x^2). Write this term above the dividing line.

3x^2
___________________
(x^2 - 8) | 3x^4 - 2x^3 + 16x - 192

Step 3: Multiply the entire divisor (x^2 - 8) by the first term of the quotient (3x^2). Write the result under the dividend, and subtract it.

3x^2
___________________
(x^2 - 8) | 3x^4 - 2x^3 + 16x - 192
- (3x^4 - 24x^2)

3x^2
___________________
(x^2 - 8) | - 2x^3 + 16x - 192
- ( - 2x^3 + 16x)
_________________
- 192

Step 4: Bring down the next term from the dividend (in this case, -192) and repeat the process.

3x^2 - 24
___________________
(x^2 - 8) | 3x^4 - 2x^3 + 16x - 192
- (3x^4 - 24x^2)
_________________
0 - 40x^2 + 16x - 192
- ( - 40x^2 + 320)
___________________
- 512

Step 5: Since there is no remainder after the subtraction, the final quotient is 3x^2 - 24.

Therefore, the simplified expression is:
(3x^4 - 2x^3 + 16x - 192) / (x^2 - 8) = 3x^2 - 24.

I hope this explanation helps you understand how to divide and simplify polynomial expressions using polynomial long division. Let me know if you have any further questions!