6x^3+4x^2-2x+5 divided by 3x^2+2x+1
To divide the polynomial (6x^3 + 4x^2 - 2x + 5) by (3x^2 + 2x + 1), you can use long division.
```
2x^2 - x + 4
--------------------------
3x^2 + 2x + 1 | 6x^3 + 4x^2 - 2x + 5
- (6x^3 + 4x^2 + 2x)
----------------------
-4x + 5
- (-4x - 2)
---------------
7
```
The quotient is 2x^2 - x + 4 and the remainder is 7.
To divide the polynomial (6x^3 + 4x^2 - 2x + 5) by the polynomial (3x^2 + 2x + 1), you can use polynomial long division. Here's how you can do it:
Step 1: Make sure the polynomials are written in descending order of powers of x. If any power is missing, insert a placeholder with a coefficient of 0.
(6x^3 + 4x^2 - 2x + 5) ÷ (3x^2 + 2x + 1)
Step 2: Begin the process of long division by dividing the highest power term of the dividend (6x^3) by the highest power term of the divisor (3x^2). The result is 2x. Write this above the division bar.
2x
__
3x^2 + 2x + 1
Step 3: Multiply the entire divisor (3x^2 + 2x + 1) by the result obtained in Step 2 (2x). Write the result below the dividend, and then subtract it from the original dividend.
2x
__
3x^2 + 2x + 1 Dividend
- (2x^3 + 4x^2 + 2x) Result × Divisor
Resulting in:
____________
1x^2 - 4x + 5 Remainder
Step 4: Bring down the next term from the original dividend (in this case, -2x), and repeat Steps 2 and 3.
2x -4
__
3x^2 + 2x + 1 Dividend
-(2x^3 + 4x^2 + 2x) Result × Divisor
____________
- 2x^2 + 2x
2x^2 + 4x + 2 Result × Divisor
Resulting in:
______________
1x^2 - 4x + 5
2x^2 - 2x Remainder
Step 5: Repeat Steps 2 and 3 with the remainder obtained in Step 4 (-2x).
2x - 4 + (-2)
__
3x^2 + 2x + 1 Dividend
-(2x^3 + 4x^2 + 2x) Result × Divisor
____________
- 2x^2 + 2x
2x^2 - 2x Remainder
___________
3
Resulting in:
______________
1x^2 - 4x + 5
2x^2 - 2x
3 Remainder
The final result is:
(1x^2 - 4x + 5) + (2x^2 - 2x)/(3x^2 + 2x + 1) + 3/(3x^2 + 2x + 1)
So, the quotient is 2x - 4 with a remainder of 3/(3x^2 + 2x + 1).
To divide the polynomial (6x^3 + 4x^2 - 2x + 5) by the polynomial (3x^2 + 2x + 1), follow these steps:
Step 1: Perform long division with the leading terms.
Divide the first term of the dividend (6x^3) by the first term of the divisor (3x^2), which gives you 2x.
2x
___________
3x^2 + 2x + 1
| 6x^3 + 4x^2 - 2x + 5
Step 2: Multiply the divisor by the result from step 1.
Multiply (2x) by (3x^2 + 2x + 1) to get (6x^3 + 4x^2 + 2x).
2x
___________
3x^2 + 2x + 1
| 6x^3 + 4x^2 - 2x + 5
- (6x^3 + 4x^2 + 2x)
Step 3: Subtract the result from step 2 from the dividend.
Subtract (6x^3 + 4x^2 + 2x) from (6x^3 + 4x^2 - 2x + 5) to get (-4x - 2x + 5).
2x - 6
___________
3x^2 + 2x + 1
| 6x^3 + 4x^2 - 2x + 5
- (6x^3 + 4x^2 + 2x)
___________________
-6x + 5
Step 4: Bring down the next term.
Bring down the next term from the dividend, which is (-6x + 5).
2x - 6
___________
3x^2 + 2x + 1
| 6x^3 + 4x^2 - 2x + 5
- (6x^3 + 4x^2 + 2x)
___________________
-6x + 5
Step 5: Repeat steps 1-4 until you have no more terms to bring down.
Repeat steps 1-4 using the new expression (-6x + 5).
2x - 6
___________
3x^2 + 2x + 1
| 6x^3 + 4x^2 - 2x + 5
- (6x^3 + 4x^2 + 2x)
___________________
-6x + 5
- (-6x - 4)
___________________
9
Step 6: Write the final quotient and remainder.
The final quotient is (2x - 6) and the remainder is 9.
Therefore, the polynomial (6x^3 + 4x^2 - 2x + 5) divided by (3x^2 + 2x + 1) is equal to (2x - 6) with a remainder of 9.