How do you divide #(4x^4 + 3x^3 + 3x - 2) / ( x^2 - 3 ) # using polynomial long division?

To divide (4x^4 + 3x^3 + 3x - 2) by (x^2 - 3) using polynomial long division, follow these steps:

1. Arrange the terms in descending order of degree for both the dividend and divisor. Dividend: 4x^4 + 3x^3 + 0x^2 + 3x - 2 Divisor: x^2 - 3

2. Divide the first term of the dividend (4x^4) by the first term of the divisor (x^2). The result is 4x^2.

3. Multiply the divisor (x^2 - 3) by the result obtained in step 2 (4x^2). The product is 4x^4 - 12x^2.

4. Subtract the product obtained in step 3 from the dividend. (4x^4 + 3x^3 + 3x - 2) - (4x^4 - 12x^2) = 3x^3 + 12x^2 + 3x - 2

5. Bring down the next term from the dividend. In this case, it is 3x.

6. Divide the first term of the new dividend (3x^3) by the first term of the divisor (x^2). The result is 3x.

7. Multiply the divisor (x^2 - 3) by the result obtained in step 6 (3x). The product is 3x^3 - 9x.

8. Subtract the product obtained in step 7 from the new dividend. (3x^3 + 12x^2 + 3x - 2) - (3x^3 - 9x) = 12x^2 + 12x - 2

9. Bring down the next term from the dividend. In this case, it is -2.

10. Divide the first term of the new dividend (12x^2) by the first term of the divisor (x^2). The result is 12.

11. Multiply the divisor (x^2 - 3) by the result obtained in step 10 (12). The product is 12x^2 - 36.

12. Subtract the product obtained in step 11 from the new dividend. (12x^2 + 12x - 2) - (12x^2 - 36) = 12x + 34

13. Bring down any remaining terms from the dividend. In this case, there are no more terms.

14. The quotient is the sum of the results obtained in steps 2, 6, and 10: 4x^2 + 3x + 12.

15. The remainder is the result obtained in step 13: 12x + 34.

Therefore, (4x^4 + 3x^3 + 3x - 2) divided by (x^2 - 3) using polynomial long division is equal to 4x^2 + 3x + 12 with a remainder of 12x + 34.