(x^4+3x^3-x^2+5x-3)/(x^2+1) solve using long division
@@@@@@@@__x^2__+3x__-2_______________
x^2 + 1 | x^4 +3x^3-1x^2+5x-3
@@@@@@@@@ x^4 +0x^3+1x^2
@@@@@@@@@--------------------
@@@@@@@@@@@@@@ 3x^3-2x^2+5x-3
@@@@@@@@@@@@@@ 3x^3+0x^2+3x
@@@@@@@@@@@@@@ ---------------
@@@@@@@@@@@@@@@@@@@ -2x^2+2x-3
@@@@@@@@@@@@@@@@@@@ -2x^2+0x-2
@@@@@@@@@@@@@@@@@@@@@@@@@------
@@@@@@@@@@@@@@@@@@@@@@@@@ R=2x-1
If you enter your polynomials at
http://calc101.com/webMathematica/long-divide.jsp
you can see all the details, with a little less noise.
To solve the division problem, you can use long division. Here's a step-by-step guide on how to do it:
Step 1: Write the division problem in the long division format:
______________________
x^2 + 1 | x^4 + 3x^3 - x^2 + 5x - 3
Step 2: Start with the first term of the dividend (x^4) and divide it by the first term of the divisor (x^2). The result becomes the first term of the quotient:
x^2
______________________
x^2 + 1 | x^4 + 3x^3 - x^2 + 5x - 3
x^4
- x^4 + x^2
Step 3: Multiply the first term of the quotient (x^2) by the entire divisor (x^2 + 1):
x^2
______________________
x^2 + 1 | x^4 + 3x^3 - x^2 + 5x - 3
x^4
- x^4 + x^2
_______________
0 + x^2
Step 4: Subtract the result obtained in Step 3 from the corresponding terms in the dividend. Bring down the next term (3x^3) from the dividend:
x^2
______________________
x^2 + 1 | x^4 + 3x^3 - x^2 + 5x - 3
x^4
- x^4 + x^2
_______________
0 + x^2
x^2
- 3x^2 + 3x^3
Step 5: Repeat Steps 3 and 4 for the remaining terms, taking each term of the dividend one at a time:
x^2 + 3x
______________________
x^2 + 1 | x^4 + 3x^3 - x^2 + 5x - 3
x^4
- x^4 + x^2
_______________
0 + x^2
x^2
- 3x^2 + 3x^3
_______________
0 - 3x^2 + 5x
x^2 + 3x
- (-x^2 - x)
_______________
0 - 3x^2 + 5x
0 + 4x - 3
x^2 + 3x - 4
- (x^2 + 1)
_______________
0 - 3x^2 + 5x
0 + 4x - 3
______________
4x - 2
Step 6: Once you've gone through all the terms and the remainder is less than or equal to the divisor, the division is complete.
Therefore, the quotient is x^2 + 3x - 4 and the remainder is 4x - 2.
So, the result of dividing (x^4 + 3x^3 - x^2 + 5x - 3) by (x^2 + 1) using long division is (x^2 + 3x - 4) with a remainder of (4x - 2).