Question
Long division polynomial x^2+4 / 2x^4-x^3+9x^2
Answers
GPT 3.5
To divide the polynomial x^2 + 4 by the polynomial 2x^4 - x^3 + 9x^2 using long division, follow these steps:
1. Write the dividend (x^2 + 4) and the divisor (2x^4 - x^3 + 9x^2) in descending order based on the powers of x:
Dividend: 0x^4 + 9x^2 + x^2 + 4
Divisor: 2x^4 - x^3 + 9x^2
2. Divide the leading term of the dividend (0x^4) by the leading term of the divisor (2x^4). The quotient is 0:
0x^4 / 2x^4 = 0
3. Multiply the quotient (0) by the divisor (2x^4 - x^3 + 9x^2):
0 * (2x^4 - x^3 + 9x^2) = 0
4. Subtract the result from step 3 from the dividend:
(0x^4 + 9x^2 + x^2 + 4) - 0 = 0x^4 + 10x^2 + x^2 + 4
5. Bring down the next term from the dividend (10x^2):
Remainder: 0x^4 + 10x^2 + x^2 + 4
6. Repeat steps 2-5 until all the terms have been processed:
- Divide the leading term of the remainder (0x^4) by the leading term of the divisor (2x^4). The quotient is 0.
- Multiply the quotient (0) by the divisor (2x^4 - x^3 + 9x^2). Result: 0.
- Subtract the result from step 7 from the remainder: (0x^4 + 10x^2 + x^2 + 4) - 0 = 0x^4 + 11x^2 + x^2 + 4.
- Bring down the next term from the dividend (11x^2): 0x^4 + 11x^2 + x^2 + 4.
- Divide the leading term of the new remainder (0x^4) by the leading term of the divisor (2x^4). The quotient is 0.
- Multiply the quotient (0) by the divisor (2x^4 - x^3 + 9x^2). Result: 0.
- Subtract the result from step 9 from the new remainder: (0x^4 + 11x^2 + x^2 + 4) - 0 = 0x^4 + 12x^2 + x^2 + 4.
- Bring down the next term from the dividend (12x^2): 0x^4 + 12x^2 + x^2 + 4.
7. Since the degree of the remainder (0x^4 + 12x^2 + x^2 + 4) is lower than the degree of the divisor (2x^4 - x^3 + 9x^2), the long division process stops.
The final result of the division of x^2 + 4 by 2x^4 - x^3 + 9x^2 is:
Quotient: 0
Remainder: 0x^4 + 12x^2 + x^2 + 4 (which can also be written as x^2 + 12x^2 + 4)
1. Write the dividend (x^2 + 4) and the divisor (2x^4 - x^3 + 9x^2) in descending order based on the powers of x:
Dividend: 0x^4 + 9x^2 + x^2 + 4
Divisor: 2x^4 - x^3 + 9x^2
2. Divide the leading term of the dividend (0x^4) by the leading term of the divisor (2x^4). The quotient is 0:
0x^4 / 2x^4 = 0
3. Multiply the quotient (0) by the divisor (2x^4 - x^3 + 9x^2):
0 * (2x^4 - x^3 + 9x^2) = 0
4. Subtract the result from step 3 from the dividend:
(0x^4 + 9x^2 + x^2 + 4) - 0 = 0x^4 + 10x^2 + x^2 + 4
5. Bring down the next term from the dividend (10x^2):
Remainder: 0x^4 + 10x^2 + x^2 + 4
6. Repeat steps 2-5 until all the terms have been processed:
- Divide the leading term of the remainder (0x^4) by the leading term of the divisor (2x^4). The quotient is 0.
- Multiply the quotient (0) by the divisor (2x^4 - x^3 + 9x^2). Result: 0.
- Subtract the result from step 7 from the remainder: (0x^4 + 10x^2 + x^2 + 4) - 0 = 0x^4 + 11x^2 + x^2 + 4.
- Bring down the next term from the dividend (11x^2): 0x^4 + 11x^2 + x^2 + 4.
- Divide the leading term of the new remainder (0x^4) by the leading term of the divisor (2x^4). The quotient is 0.
- Multiply the quotient (0) by the divisor (2x^4 - x^3 + 9x^2). Result: 0.
- Subtract the result from step 9 from the new remainder: (0x^4 + 11x^2 + x^2 + 4) - 0 = 0x^4 + 12x^2 + x^2 + 4.
- Bring down the next term from the dividend (12x^2): 0x^4 + 12x^2 + x^2 + 4.
7. Since the degree of the remainder (0x^4 + 12x^2 + x^2 + 4) is lower than the degree of the divisor (2x^4 - x^3 + 9x^2), the long division process stops.
The final result of the division of x^2 + 4 by 2x^4 - x^3 + 9x^2 is:
Quotient: 0
Remainder: 0x^4 + 12x^2 + x^2 + 4 (which can also be written as x^2 + 12x^2 + 4)