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(5x4−2x3−7x2−39)÷(x2+2x−4)

5 min read Jun 16, 2024
(5x4−2x3−7x2−39)÷(x2+2x−4)

Polynomial Long Division: (5x⁴−2x³−7x²−39)÷(x²+2x−4)

This article will walk through the process of dividing the polynomial (5x⁴−2x³−7x²−39) by (x²+2x−4) using polynomial long division.

Understanding Polynomial Long Division

Polynomial long division is similar to the long division you learned in elementary school, but instead of working with numbers, you're working with polynomials.

Here are the key steps:

  1. Set up the division problem. Write the dividend (the polynomial being divided) inside the division symbol and the divisor (the polynomial dividing) outside.
  2. Focus on the leading terms. Divide the leading term of the dividend by the leading term of the divisor. This will be the first term of the quotient.
  3. Multiply the divisor by the first term of the quotient. Write the result below the dividend, aligning terms with the same degree.
  4. Subtract the result. Change the signs of the terms in the product and add them to the dividend.
  5. Bring down the next term. Repeat steps 2-4 with the new polynomial until the degree of the remaining polynomial is less than the degree of the divisor.

Let's divide!

  1. Set up:

         ____________
x²+2x-4 | 5x⁴ - 2x³ - 7x² - 39

  2. Divide leading terms:

    • The leading term of the dividend (5x⁴) divided by the leading term of the divisor (x²) is 5x².

  3. Multiply and subtract:

    • Multiply the divisor (x²+2x-4) by 5x²: 5x⁴ + 10x³ - 20x²
    • Subtract this result from the dividend:
         ____________

    x²+2x-4 | 5x⁴ - 2x³ - 7x² - 39 -(5x⁴ + 10x³ - 20x²) ------------------- -12x³ + 13x² - 39

  4. Bring down the next term:

    • Bring down the -39:
        ____________
x²+2x-4 | 5x⁴ - 2x³ - 7x² - 39
         -(5x⁴ + 10x³ - 20x²)
         -------------------
                -12x³ + 13x² - 39

  5. Repeat steps 2-4:

    • Divide the new leading term (-12x³) by the leading term of the divisor (x²): -12x.
    • Multiply the divisor (x²+2x-4) by -12x: -12x³ - 24x² + 48x
    • Subtract:
         ____________

    x²+2x-4 | 5x⁴ - 2x³ - 7x² - 39 -(5x⁴ + 10x³ - 20x²) ------------------- -12x³ + 13x² - 39 -(-12x³ - 24x² + 48x) ------------------------- 37x² - 48x - 39

  6. Repeat again:

    • Divide the new leading term (37x²) by the leading term of the divisor (x²): 37.
    • Multiply the divisor (x²+2x-4) by 37: 37x² + 74x - 148
    • Subtract:
         ____________

    x²+2x-4 | 5x⁴ - 2x³ - 7x² - 39 -(5x⁴ + 10x³ - 20x²) ------------------- -12x³ + 13x² - 39 -(-12x³ - 24x² + 48x) ------------------------- 37x² - 48x - 39 -(37x² + 74x - 148) ------------------------ -122x + 109

  7. The remainder:

    • The degree of the remaining polynomial (-122x + 109) is less than the degree of the divisor (x²+2x-4), so we stop here.

Final Result:

The quotient is 5x² - 12x + 37 and the remainder is -122x + 109. Therefore, we can write the result as:

(5x⁴−2x³−7x²−39) ÷ (x²+2x−4) = 5x² - 12x + 37 + (-122x + 109) / (x²+2x−4)

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