%2F(x%5E2-x%2B1).jpeg "(x^4+2x^2-x+5)/(x^2-x+1)")
Polynomial Long Division: (x^4+2x^2-x+5)/(x^2-x+1)
This article will walk you through the process of performing polynomial long division to simplify the expression (x^4+2x^2-x+5)/(x^2-x+1).
Step 1: Set Up the Division
Start by setting up the division problem in the standard long division format.
__________
x^2-x+1 | x^4 + 0x^3 + 2x^2 - x + 5
Notice we included a 0x^3 term as a placeholder for the missing cubic term in the dividend.
Step 2: Divide the Leading Terms
Focus on the leading terms of the divisor (x^2) and the dividend (x^4). Ask yourself: "What do I multiply x^2 by to get x^4?" The answer is x^2.
Write x^2 above the x^4 term in the quotient.
x^2
x^2-x+1 | x^4 + 0x^3 + 2x^2 - x + 5
Step 3: Multiply and Subtract
Multiply the entire divisor (x^2-x+1) by the term you just placed in the quotient (x^2). This gives you:
x^2
x^2-x+1 | x^4 + 0x^3 + 2x^2 - x + 5
x^4 - x^3 + x^2
Now, subtract this entire expression from the dividend. Remember to change the signs when subtracting!
x^2
x^2-x+1 | x^4 + 0x^3 + 2x^2 - x + 5
x^4 - x^3 + x^2
------------------
x^3 + x^2 - x
Step 4: Bring Down the Next Term
Bring down the next term from the dividend (-x).
x^2
x^2-x+1 | x^4 + 0x^3 + 2x^2 - x + 5
x^4 - x^3 + x^2
------------------
x^3 + x^2 - x + 5
Step 5: Repeat the Process
Now, focus on the new leading terms: x^3 and x^2. Ask yourself: "What do I multiply x^2 by to get x^3?" The answer is x.
Write x in the quotient next to x^2.
x^2 + x
x^2-x+1 | x^4 + 0x^3 + 2x^2 - x + 5
x^4 - x^3 + x^2
------------------
x^3 + x^2 - x + 5
Multiply the divisor by x and subtract.
x^2 + x
x^2-x+1 | x^4 + 0x^3 + 2x^2 - x + 5
x^4 - x^3 + x^2
------------------
x^3 + x^2 - x + 5
x^3 - x^2 + x
--------------
2x^2 - 2x + 5
Step 6: Continue until the Degree of the Remainder is Less than the Divisor
Repeat the process of dividing the leading terms, multiplying, and subtracting.
x^2 + x + 2
x^2-x+1 | x^4 + 0x^3 + 2x^2 - x + 5
x^4 - x^3 + x^2
------------------
x^3 + x^2 - x + 5
x^3 - x^2 + x
--------------
2x^2 - 2x + 5
2x^2 - 2x + 2
-------------
3
We stop here because the degree of the remainder (0) is less than the degree of the divisor (2).
Solution
The result of the division is:
(x^4+2x^2-x+5)/(x^2-x+1) = x^2 + x + 2 + 3/(x^2-x+1)