(x3+x2+x+2)÷(x2−1) Long Division

Jasonbradley

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Jun 17, 2024

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Long Division of Polynomials: (x³ + x² + x + 2) ÷ (x² - 1)

Long division of polynomials is a process used to divide one polynomial by another. This process is similar to the long division of numbers, with some key differences. Let's walk through the steps of dividing (x³ + x² + x + 2) by (x² - 1):

1. Set up the Division:

Write the dividend (x³ + x² + x + 2) inside the division symbol and the divisor (x² - 1) outside.

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

2. Divide the Leading Terms:

• Divide the leading term of the dividend (x³) by the leading term of the divisor (x²). This gives us x.
• Write x above the division symbol, aligned with the x³ term.

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

3. Multiply and Subtract:

• Multiply the divisor (x² - 1) by the term you just wrote (x). This gives us x³ - x.
• Write this result below the dividend, aligning terms with matching exponents.
• Subtract the result from the dividend.

          x      
x² - 1 | x³ + x² + x + 2 
        -(x³ - x)
        _________
              x² + 2x + 2 

4. Bring Down the Next Term:

• Bring down the next term of the dividend (+2) to the bottom row.

          x      
x² - 1 | x³ + x² + x + 2 
        -(x³ - x)
        _________
              x² + 2x + 2 

5. Repeat Steps 2-4:

• Divide the new leading term (x²) by the leading term of the divisor (x²). This gives us 1.
• Write 1 next to the x in the quotient.
• Multiply the divisor (x² - 1) by 1 and subtract the result.

          x + 1   
x² - 1 | x³ + x² + x + 2 
        -(x³ - x)
        _________
              x² + 2x + 2 
              -(x² - 1)
              _________
                     2x + 3

6. Determine the Remainder:

• The degree of the remaining polynomial (2x + 3) is less than the degree of the divisor (x² - 1). Therefore, we have reached our final remainder.

7. Express the Result:

The result of the division is:

(x³ + x² + x + 2) ÷ (x² - 1) = x + 1 + (2x + 3)/(x² - 1)

This means that the quotient is x + 1, and the remainder is 2x + 3.

Key Points to Remember:

• The process continues until the degree of the remainder is less than the degree of the divisor.
• If the remainder is zero, the divisor is a factor of the dividend.
• Long division can be used to factor polynomials, find the roots of a polynomial, and simplify expressions involving polynomials.