Just checking to see if I got this right.
Thanks,
(3x^4--8x^2 -3x-3) / (x^2-3)
3x^2+1 remainder -3x
3x^2 + 1 + (-3x)/(x^2 - 3)
Yes, your answer is correct, but I was taught that the remainder is expressed as,
(-3x)/(x^2 - 3)
Great! Let's go through the steps to solve the division problem to confirm your answer.
1. Begin by dividing the highest degree term of the numerator, which is 3x^4, by the highest degree term of the denominator, x^2. This gives us 3x^(4-2) = 3x^2.
2. Now, multiply this result (3x^2) by the entire denominator, (x^2-3):
3x^2 * (x^2-3) = 3x^4 - 9x^2.
3. Next, subtract this result (3x^4 - 9x^2) from the numerator (3x^4-8x^2 -3x-3):
(3x^4 - 8x^2 - 3x - 3) - (3x^4 - 9x^2) = -8x^2 - 3x - 3.
4. This gives us the updated numerator: -8x^2 - 3x - 3.
5. We continue the division process by dividing the new highest degree term, -8x^2, by the highest degree term of the denominator, x^2, which is -8x^2 / x^2 = -8.
6. Multiply this result (-8) by the entire denominator (x^2-3):
-8 * (x^2-3) = -8x^2 + 24.
7. Subtract this result (-8x^2 + 24) from the updated numerator (-8x^2 - 3x - 3):
(-8x^2 - 3x - 3) - (-8x^2 + 24) = -3x - 27.
8. The updated numerator is now -3x - 27.
9. Repeat the process by dividing the new highest degree term, -3x, by the highest degree term of the denominator, x^2, which is -3x / x^2 = -3/x.
10. Multiply this result (-3/x) by the entire denominator (x^2-3):
(-3/x) * (x^2-3) = -3x + 9/x.
11. Subtract this result (-3x + 9/x) from the updated numerator (-3x - 27):
(-3x - 27) - (-3x + 9/x) = -27 - 9/x.
12. The remainder in this step is -27 - 9/x.
Therefore, the final result after the division is 3x^2 + 1 with a remainder of -27 - 9/x, or simply -3x - 27. So, it seems like your answer of 3x^2 + 1 remainder -3x is incorrect.