Finnd the quotient and remainder when F(x)=6x^3+4x^2-x+5 divided by 3x^2+2x+1 using the long division method
Let's perform the long division:
2x + 1
_______________
3x^2+2x+1 | 6x^3+4x^2-x+5
To start the division, we divide the first term of the dividend by the first term of the divisor:
2x + 1
____________________
3x^2+2x+1 | 6x^3+4x^2-x+5
- (6x^3 + 4x^2 + 2x^2)
________________________
-x+5
Now, we bring down the next term, -x, and divide it by the first term of the divisor:
2x + 1
_____________________
3x^2+2x+1 | 6x^3+4x^2-x+5
- (6x^3 + 4x^2 + 2x^2)
________________________
-x+ 5
-(-x+0)
__________________
5
The quotient is 2x + 1 and the remainder is 5.
To find the quotient and remainder when dividing the polynomial F(x) = 6x^3 + 4x^2 - x + 5 by 3x^2 + 2x + 1 using long division, follow these steps:
Step 1: Write the polynomials in descending order of their powers.
F(x) = 6x^3 + 4x^2 - x + 5
Divisor: D(x) = 3x^2 + 2x + 1
Step 2: Divide the highest power term of F(x) by the highest power term of D(x) and write the result as the first term of the quotient.
The highest power term of F(x) is 6x^3, and the highest power term of D(x) is 3x^2. So, the first term of the quotient is (6x^3 / 3x^2) = 2x.
Step 3: Multiply the divisor by the first term of the quotient, and subtract the result from the dividend.
Multiply 2x by D(x): 2x * (3x^2 + 2x + 1) = 6x^3 + 4x^2 + 2x
Subtract this result from F(x):
6x^3 + 4x^2 - x + 5 - (6x^3 + 4x^2 + 2x) = -3x - 5.
Step 4: Bring down the next term from the dividend.
The next term to bring down is -3x.
Step 5: Divide the newly formed expression (-3x - 5) by the divisor (3x^2 + 2x + 1) and repeat step 2, 3, and 4 until there are no more terms left in the dividend.
-3x / 3x^2 = (-1/x)
(-1/x) * D(x) = (-1/x) * (3x^2 + 2x + 1) = (-3x - 2 - 1/x)
Now, subtract this result from -3x - 5:
(-3x - 5) - (-3x - 2 - 1/x) = -3
Since the result of this subtraction is constant (-3), there are no further terms to bring down.
Step 6: The quotient is the sum of all the terms obtained from the division.
The quotient is 2x - (1/x) - 3.
Step 7: The remainder is the last term obtained from the division.
The remainder is -3.
Therefore, the quotient is 2x - (1/x) - 3, and the remainder is -3 when dividing F(x) by D(x) using long division.
To find the quotient and remainder when dividing the polynomial F(x) = 6x^3 + 4x^2 - x + 5 by the polynomial 3x^2 + 2x + 1 using long division, follow the steps below:
Step 1: Write the dividend and divisor in descending order of powers of x.
```
2x^2
___________
3x^2 + 2x + 1 | 6x^3 + 4x^2 - x + 5
```
Step 2: Divide the leading term of the dividend (6x^3) by the leading term of the divisor (3x^2). The result is the first term of the quotient.
```
2x^2
___________
3x^2 + 2x + 1 | 6x^3 + 4x^2 - x + 5
-2x^2
```
Step 3: Multiply the divisor (3x^2 + 2x + 1) by the first term of the quotient (-2x^2). Write the product under the dividend, aligning it with the corresponding term.
```
2x^2
___________
3x^2 + 2x + 1 | 6x^3 + 4x^2 - x + 5
-2x^2 - 4x^2 - 2x
```
Step 4: Subtract the product from the dividend.
```
2x^2
___________
3x^2 + 2x + 1 | 6x^3 + 4x^2 - x + 5
-2x^2 - 4x^2 - 2x
-----------------
2x + x + 5
```
Step 5: Bring down the next term from the dividend (x).
```
2x^2 + x
___________
3x^2 + 2x + 1 | 6x^3 + 4x^2 - x + 5
-2x^2 - 4x^2 - 2x
-----------------
2x + x + 5
- (2x + x)
```
Step 6: Divide the leading term of the new dividend (2x) by the leading term of the divisor (3x^2). The result is the next term of the quotient.
```
2x^2 + x
___________
3x^2 + 2x + 1 | 6x^3 + 4x^2 - x + 5
-2x^2 - 4x^2 - 2x
-----------------
2x + x + 5
- (2x + x)
-2x
```
Step 7: Multiply the divisor (3x^2 + 2x + 1) by the next term of the quotient (-2x). Write the product under the new dividend, aligning it with the corresponding term.
```
2x^2 + x
___________
3x^2 + 2x + 1 | 6x^3 + 4x^2 - x + 5
-2x^2 - 4x^2 - 2x
-----------------
2x + x + 5
- (2x + x)
-2x - 4x^2 -2
```
Step 8: Subtract the product from the new dividend.
```
2x^2 + x
___________
3x^2 + 2x + 1 | 6x^3 + 4x^2 - x + 5
-2x^2 - 4x^2 - 2x
-----------------
2x + x + 5
- (2x + x)
-2x - 4x^2 - 2
-------------
4x^2 - 3x + 5
```
Step 9: The new dividend (4x^2 - 3x + 5) is the remainder.
Therefore, the quotient is 2x^2 + x - 2, and the remainder is 4x^2 - 3x + 5.