Divide using long division or synthetic division.
(21x^3 - 7)/(3x - 1)
This is what I have so far, but I am having a hard time with what to do next.
3x-1/ 7(3x^3-1)
horrible division!
it is hard to show long division on here, but the answer I got was
(21x^3 - 7)/(3x - 1) = (7x^2 + (7/3)x + 7/9) - 56/(9(3x-1))
To divide the polynomial (21x^3 - 7) by (3x - 1) using long division, follow these steps:
1. Write down the dividend (21x^3 - 7) and the divisor (3x - 1).
2. Divide the first term of the dividend by the first term of the divisor. In this case, divide 21x^3 by 3x, which gives you 7x^2.
3. Write the result (7x^2) above the division bar, aligned with the dividend.
4. Multiply the divisor (3x - 1) by the result obtained in step 2 (7x^2). This gives you 21x^3 - 7x^2.
5. Subtract the result obtained in step 4 (21x^3 - 7x^2) from the dividend (21x^3 - 7).
- (21x^3 - 7) - (21x^3 - 7x^2) = 0 + 7x^2 = 7x^2.
6. Bring down the next term from the dividend, which is -7.
7. Now you have the new dividend as 7x^2 - 7. Repeat steps 2 to 7 until there are no more terms to bring down or until the degree of the new dividend is less than the degree of the divisor.
Repeat steps 2 to 7 with the new dividend (7x^2 - 7):
8. Divide the first term of the new dividend (7x^2) by the first term of the divisor (3x). The result is 2x.
9. Write the result (2x) above the division bar, aligned with the previous result.
10. Multiply the divisor (3x - 1) by the result obtained in step 8 (2x), giving you 6x^2 - 2x.
11. Subtract the result obtained in step 10 (6x^2 - 2x) from the new dividend (7x^2 - 7).
- (7x^2 - 7) - (6x^2 - 2x) = x^2 - 2x - 7.
12. Bring down the next term from the dividend, which is -7.
13. Now you have the new dividend as x^2 - 2x - 7. Continue repeating steps 2 to 7 until there are no more terms to bring down or until the degree of the new dividend is less than the degree of the divisor.
Repeat steps 2 to 7 with the new dividend (x^2 - 2x - 7):
14. Divide the first term of the new dividend (x^2) by the first term of the divisor (3x). The result is (1/3)x.
15. Write the result ((1/3)x) above the division bar, aligned with the previous results.
16. Multiply the divisor (3x - 1) by the result obtained in step 14 ((1/3)x), giving you (1/3)x^2 - (1/3)x.
17. Subtract the result obtained in step 16 ((1/3)x^2 - (1/3)x) from the new dividend (x^2 - 2x - 7).
- (x^2 - 2x - 7) - ((1/3)x^2 - (1/3)x) = (2/3)x^2 - (5/3)x - 7.
18. There are no more terms to bring down, and the degree of the new dividend ((2/3)x^2 - (5/3)x - 7) is less than the degree of the divisor (3x - 1), so the division is complete.
The quotient is 7x^2 + 2x + (1/3), and the remainder is (2/3)x^2 - (5/3)x - 7.