(x3+x2+3x−4)÷(x2+2x+1)

3 min read Jun 17, 2024
(x3+x2+3x−4)÷(x2+2x+1)

Dividing Polynomials: (x³ + x² + 3x - 4) ÷ (x² + 2x + 1)

This article will guide you through the process of dividing the polynomial (x³ + x² + 3x - 4) by (x² + 2x + 1) using long division.

Step 1: Set up the Long Division

First, write the dividend (x³ + x² + 3x - 4) inside the division symbol and the divisor (x² + 2x + 1) outside.

             ________
x² + 2x + 1 | x³ + x² + 3x - 4 

Step 2: Divide the Leading Terms

Focus on the leading terms of both the divisor and the dividend. Divide the leading term of the dividend (x³) by the leading term of the divisor (x²). This gives us x.

             x       
x² + 2x + 1 | x³ + x² + 3x - 4 

Step 3: Multiply and Subtract

Multiply the quotient (x) by the divisor (x² + 2x + 1) and write the result below the dividend.

             x       
x² + 2x + 1 | x³ + x² + 3x - 4 
             x³ + 2x² + x

Subtract the result from the dividend.

             x       
x² + 2x + 1 | x³ + x² + 3x - 4 
             x³ + 2x² + x
             ---------
                  -x² + 2x - 4

Step 4: Repeat Steps 2 and 3

Bring down the next term (-4) from the dividend. Focus on the leading term of the new dividend (-x²) and divide it by the leading term of the divisor (x²). This gives us -1.

             x - 1     
x² + 2x + 1 | x³ + x² + 3x - 4 
             x³ + 2x² + x
             ---------
                  -x² + 2x - 4
                  -x² - 2x - 1

Multiply -1 by the divisor and subtract the result.

             x - 1     
x² + 2x + 1 | x³ + x² + 3x - 4 
             x³ + 2x² + x
             ---------
                  -x² + 2x - 4
                  -x² - 2x - 1
                  ---------
                        4x - 3 

Step 5: Interpret the Result

Since the degree of the new dividend (4x - 3) is less than the degree of the divisor (x² + 2x + 1), we stop here.

Therefore, the result of dividing (x³ + x² + 3x - 4) by (x² + 2x + 1) is:

x - 1 + (4x - 3) / (x² + 2x + 1)

This means that the quotient is x - 1 and the remainder is 4x - 3.

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