Divide the polynomials using long division.
(4b^4+14b^2+71)/(2b^2+3)
2b^2 + 3
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2b^2 + 0b + 11
(4b^4 + 0b^3 + 14b^2 + 0b + 71)
- (4b^4 + 6b^2)
________________
-8b^2 + 0b + 71
+ ( -8b^2 - 12)
_______________________________
12b + 83
Therefore, (4b^4 + 14b^2 + 71) divided by (2b^2 + 3) equals (2b^2 + 11) with a remainder of (12b + 83).
so what is the answer?
The answer is:
(4b^4 + 14b^2 + 71) / (2b^2 + 3) = 2b^2 + 11 + (12b + 83) / (2b^2 + 3)
divide polynomials by long division
(6b^4-1)/(2b+1)
To divide the polynomial (6b^4 - 1) by (2b + 1) using long division, we follow these steps:
1. Set up the long division as follows:
3b^3 -3b^2 +6b -6
-------------------------------------
2b + 1 | 6b^4 0b^3 0b^2 0b -1
2. Divide the first term of the dividend (6b^4) by the first term of the divisor (2b) to get 3b^3. Place this result above the line and multiply it by the divisor.
3b^3
-------------
2b + 1 | 6b^4
3. Multiply the divisor (2b + 1) by the term we just obtained (3b^3) and write the result below the dividend, lining up similar terms:
3b^3 -3b^2 +6b -6
-------------------------------------
2b + 1 | 6b^4 0b^3 0b^2 0b -1
- (6b^4 + 3b^3)
4. Subtract the result obtained in step 3 from the dividend:
3b^3 -3b^2 +6b -6
-------------------------------------
2b + 1 | 6b^4 0b^3 0b^2 0b -1
- (6b^4 + 3b^3)
---------------------
- 3b^3
5. Bring down the next term (-1) from the dividend:
3b^3 -3b^2 +6b -6
-------------------------------------
2b + 1 | 6b^4 0b^3 0b^2 0b -1
- (6b^4 + 3b^3)
---------------------
- 3b^3
-1
6. Repeat steps 2-5 until you bring down all the terms from the dividend:
3b^3 -3b^2 +6b -6
-------------------------------------
2b + 1 | 6b^4 0b^3 0b^2 0b -1
- (6b^4 + 3b^3)
---------------------
- 3b^3
-1
------------------
-6b^2
+ 6b
----------------
0b - 7
7. Analyzing the last result (-6b^2 + 6b - 7), we can see that:
- The quotient is 3b^3 - 3b^2 + 6b - 6
- The remainder is -6b^2 + 6b - 7
Therefore, (6b^4 - 1) divided by (2b + 1) equals 3b^3 - 3b^2 + 6b - 6 with a remainder of -6b^2 + 6b - 7.
To divide the polynomials using long division, follow these steps:
Step 1: Start by writing the dividend (4b^4 + 14b^2 + 71) and the divisor (2b^2 + 3) in long division format, with the dividend as the numerator and the divisor as the denominator.
_______________
2b^2 + 3 | 4b^4 + 14b^2 + 71
Step 2: Divide the first term of the dividend by the first term of the divisor, which is (4b^4 / 2b^2). The result is 2b^2.
2b^2
_______________
2b^2 + 3 | 4b^4 + 14b^2 + 71
Step 3: Multiply the divisor by the quotient from step 2, which is 2b^2. (2b^2) * (2b^2 + 3) = 4b^4 + 6b^2.
2b^2
_______________
2b^2 + 3 | 4b^4 + 14b^2 + 71
- (4b^4 + 6b^2)
Step 4: Subtract the product obtained in step 3 from the dividend. (4b^4 + 14b^2 + 71) - (4b^4 + 6b^2) = 8b^2 + 71.
2b^2
_______________
2b^2 + 3 | 4b^4 + 14b^2 + 71
- (4b^4 + 6b^2)
_______________
8b^2 + 71
Step 5: Bring down the next term from the dividend, which is 8b^2. The new dividend becomes 8b^2 + 71.
2b^2
_______________
2b^2 + 3 | 4b^4 + 14b^2 + 71
- (4b^4 + 6b^2)
_______________
8b^2 + 71
Step 6: Divide the first term of the new dividend (8b^2) by the first term of the divisor (2b^2). The result is 4.
2b^2 + 4
_______________
2b^2 + 3 | 4b^4 + 14b^2 + 71
- (4b^4 + 6b^2)
_______________
8b^2 + 71
Step 7: Multiply the divisor by the quotient from step 6, which is 4. (4) * (2b^2 + 3) = 8b^2 + 12.
2b^2 + 4
_______________
2b^2 + 3 | 4b^4 + 14b^2 + 71
- (4b^4 + 6b^2)
_______________
8b^2 + 71
- (8b^2 + 12)
Step 8: Subtract the product obtained in step 7 from the new dividend. (8b^2 + 71) - (8b^2 + 12) = 59.
2b^2 + 4
_______________
2b^2 + 3 | 4b^4 + 14b^2 + 71
- (4b^4 + 6b^2)
_______________
8b^2 + 71
- (8b^2 + 12)
_______________
59
Step 9: Since there are no more terms to bring down, the division is complete. The quotient is 2b^2 + 4, and the remainder is 59.
Therefore, the final result of the division is:
Quotient: 2b^2 + 4
Remainder: 59