(x3+x2+x+2)÷(x2−1)=

Jasonbradley

4 min read

·

Jun 17, 2024

19

Share

Solving the Polynomial Division: (x³ + x² + x + 2) ÷ (x² - 1)

This article will walk through the steps of dividing the polynomial (x³ + x² + x + 2) by (x² - 1). We'll use the long division method to solve this problem.

Understanding Long Division with Polynomials

Long division with polynomials works similarly to long division with numbers. We aim to find a quotient polynomial that, when multiplied by the divisor (x² - 1), gives us the dividend (x³ + x² + x + 2).

Here's the breakdown of the process:

1. Set up the division:

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

2. Divide the leading terms:

   • The leading term of the dividend (x³) is divided by the leading term of the divisor (x²). This gives us x.
   • Write this quotient (x) above the dividend.

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

3. Multiply the quotient by the divisor:

   • Multiply the quotient (x) by the divisor (x² - 1). This results in x³ - x.

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

4. Subtract the result from the dividend:

   • Subtract the result (x³ - x) from the dividend. Remember to change the signs of the terms you're subtracting.

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

5. Bring down the next term:

   • Bring down the next term (+2) from the dividend.

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

6. Repeat steps 2-5:

   • Divide the new leading term (x²) by the leading term of the divisor (x²), which gives us 1.
   • Write this quotient (1) next to the first quotient (x) in the answer.
   • Multiply the new quotient (1) by the divisor (x² - 1) to get x² - 1.
   • Subtract this result from the expression below the line.

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

7. Continue until the degree of the remaining polynomial is less than the degree of the divisor.

   • The degree of the remaining polynomial (2x + 3) is 1, which is less than the degree of the divisor (x² - 1).

Final Result

Therefore, the solution to the division (x³ + x² + x + 2) ÷ (x² - 1) is:

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

This means the quotient is x + 1 and the remainder is 2x + 3. The remainder is expressed as a fraction with the divisor as the denominator.