Find the quotient and remainde when Fx=4x^3+3x^2-2x is divided by 2x+l
To find the quotient and remainder when F(x) = 4x^3 + 3x^2 - 2x is divided by 2x + 1, we can use polynomial long division.
2x^2 - x + 3
_____________________
2x + 1 | 4x^3 + 3x^2 - 2x + 0
To start, we divide the leading term of F(x) by the leading term of 2x + 1, which is 2x^2.
2x^2
__________
2x + 1 | 4x^3 + 3x^2 - 2x + 0
Next, we multiply (2x^2) by (2x + 1) to get 4x^3 + 2x^2. We subtract this from the original polynomial:
2x^2 - x + 3
_____________________
2x + 1 | 4x^3 + 3x^2 - 2x + 0
- (4x^3 + 2x^2)
________________
x^2 - 2x
We now bring down the next term, which is -2x:
2x^2 - x + 3
_____________________
2x + 1 | 4x^3 + 3x^2 - 2x + 0
- (4x^3 + 2x^2)
________________
x^2 - 2x
- (x^2 - 2x)
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0
Since we have no more terms to bring down and continue dividing, our remainder is 0. Therefore, the quotient when F(x) is divided by 2x + 1 is 2x^2 - x + 3, and the remainder is 0.
To find the quotient and remainder when the polynomial F(x) = 4x^3 + 3x^2 - 2x is divided by 2x + 1, we can use polynomial long division.
Step 1: Write the dividend (Fx) and divisor (2x + 1) in the long division format:
________
2x + 1 | 4x^3 + 3x^2 - 2x
Step 2: Divide the highest degree term of the dividend (4x^3) by the highest degree term of the divisor (2x). The result is 2x^2, so write this above the line.
2x^2
________
2x + 1 | 4x^3 + 3x^2 - 2x
Step 3: Multiply the divisor (2x + 1) by the result (2x^2) and write the product (4x^3 + 2x^2) below the dividend line.
2x^2
________
2x + 1 | 4x^3 + 3x^2 - 2x
- (4x^3 + 2x^2)
Step 4: Subtract the product from the dividend. Line up similar terms and perform the subtraction.
2x^2
________
2x + 1 | 4x^3 + 3x^2 - 2x
- (4x^3 + 2x^2)
___________________
x^2 - 2x
Step 5: Bring down the next term from the dividend (-2x) and continue the process.
2x^2 + x
________
2x + 1 | 4x^3 + 3x^2 - 2x
- (4x^3 + 2x^2)
___________________
x^2 - 2x
- (x^2 + x)
Step 6: Subtract the new product (-x^2 - x) from the result of the previous step.
2x^2 + x
________
2x + 1 | 4x^3 + 3x^2 - 2x
- (4x^3 + 2x^2)
___________________
x^2 - 2x
- (x^2 + x)
___________________
- x
Step 7: At this point, we have subtracted all the terms. The final result is the quotient.
2x^2 + x - 1
________
2x + 1 | 4x^3 + 3x^2 - 2x
- (4x^3 + 2x^2)
___________________
x^2 - 2x
- (x^2 + x)
___________________
- x
So, the quotient is 2x^2 + x - 1, and the remainder is -x.
To find the quotient and remainder when Fx = 4x^3 + 3x^2 - 2x is divided by 2x + 1, we can use polynomial long division.
Step 1: Set up the long division with 4x^3 + 3x^2 - 2x divided by 2x + 1:
___________________
2x + 1 | 4x^3 + 3x^2 - 2x
Step 2: Divide the term with the highest power in the dividend (4x^3) by the term with the highest power in the divisor (2x):
___________________
2x + 1 | 4x^3 + 3x^2 - 2x
- (2x^2 + x)
Step 3: Multiply the quotient obtained (2x^2) by the entire divisor (2x + 1):
___________________
2x + 1 | 4x^3 + 3x^2 - 2x
- (2x^2 + x)
_______________
2x^2 + x
Step 4: Subtract the result obtained in step 3 from the original dividend:
___________________
2x + 1 | 4x^3 + 3x^2 - 2x
- (2x^2 + x)
_______________
2x^2 + x
- (2x^2 + x)
Step 5: Simplify the subtraction:
___________________
2x + 1 | 4x^3 + 3x^2 - 2x
- (2x^2 + x)
_______________
2x^2 + x
- 2x^2 - x
_______________
0
Step 6: The division is complete when the subtraction results in zero.
Therefore, the quotient is 2x^2 and the remainder is 0.