(3x^3+x-11)/(x+1)

Jasonbradley

4 min read

·

Jun 16, 2024

17

Share

Polynomial Long Division: (3x^3 + x - 11) / (x + 1)

This article will walk through the process of dividing the polynomial (3x³ + x - 11) by (x + 1) using polynomial long division.

Understanding Polynomial Long Division

Polynomial long division is a method for dividing polynomials, similar to how we perform long division with numbers. It involves systematically dividing the dividend polynomial by the divisor polynomial, resulting in a quotient polynomial and a remainder polynomial.

Steps for Long Division

1. Set up the division:

   • Write the dividend (3x³ + x - 11) inside the division symbol.
   • Write the divisor (x + 1) outside the division symbol.

2. Divide the leading terms:

   • Divide the leading term of the dividend (3x³) by the leading term of the divisor (x), which gives 3x².
   • Write 3x² above the division symbol, aligning it with the x³ term.

3. Multiply the quotient term by the divisor:

   • Multiply 3x² by (x + 1), which gives 3x³ + 3x².
   • Write this result below the dividend, aligning the terms.

4. Subtract:

   • Subtract the result (3x³ + 3x²) from the dividend.
   • This gives us -2x² + x - 11.

5. Bring down the next term:

   • Bring down the next term from the dividend, which is -11.
   • Now we have -2x² + x - 11.

6. Repeat steps 2-5:

   • Divide the leading term of the new dividend (-2x²) by the leading term of the divisor (x), which gives -2x.
   • Write -2x above the division symbol, aligning it with the x² term.
   • Multiply -2x by (x + 1), resulting in -2x² - 2x.
   • Subtract this from the current dividend, leaving 3x - 11.
   • Bring down the next term (which doesn't exist in this case).

7. Repeat steps 2-5 again:

   • Divide the leading term of the new dividend (3x) by the leading term of the divisor (x), which gives 3.
   • Write 3 above the division symbol, aligning it with the constant term.
   • Multiply 3 by (x + 1), resulting in 3x + 3.
   • Subtract this from the current dividend, leaving -14.

8. The remainder:

   • -14 is the remainder, as it is a constant term and cannot be divided further by (x + 1).

Result

Therefore, the result of dividing (3x³ + x - 11) by (x + 1) is:

(3x³ + x - 11) / (x + 1) = 3x² - 2x + 3 - 14/(x + 1)

This means:

• The quotient polynomial is 3x² - 2x + 3
• The remainder polynomial is -14

We can verify this by multiplying the quotient by the divisor and adding the remainder:

(3x² - 2x + 3)(x + 1) + (-14) = 3x³ + x - 11

Which confirms our result.