(x^4+2x^2-x+5)/(x^2-x+1)

5 min read Jun 17, 2024
(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)