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:
 Set up the division problem. Write the dividend (the polynomial being divided) inside the division symbol and the divisor (the polynomial dividing) outside.
 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.
 Multiply the divisor by the first term of the quotient. Write the result below the dividend, aligning terms with the same degree.
 Subtract the result. Change the signs of the terms in the product and add them to the dividend.
 Bring down the next term. Repeat steps 24 with the new polynomial until the degree of the remaining polynomial is less than the degree of the divisor.
Let's divide!

Set up:
____________ x²+2x4  5x⁴  2x³  7x²  39

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

Multiply and subtract:
 Multiply the divisor (x²+2x4) by 5x²: 5x⁴ + 10x³  20x²
 Subtract this result from the dividend:
____________
x²+2x4  5x⁴  2x³  7x²  39 (5x⁴ + 10x³  20x²)  12x³ + 13x²  39

Bring down the next term:
 Bring down the 39:
____________ x²+2x4  5x⁴  2x³  7x²  39 (5x⁴ + 10x³  20x²)  12x³ + 13x²  39

Repeat steps 24:
 Divide the new leading term (12x³) by the leading term of the divisor (x²): 12x.
 Multiply the divisor (x²+2x4) by 12x: 12x³  24x² + 48x
 Subtract:
____________
x²+2x4  5x⁴  2x³  7x²  39 (5x⁴ + 10x³  20x²)  12x³ + 13x²  39 (12x³  24x² + 48x)  37x²  48x  39

Repeat again:
 Divide the new leading term (37x²) by the leading term of the divisor (x²): 37.
 Multiply the divisor (x²+2x4) by 37: 37x² + 74x  148
 Subtract:
____________
x²+2x4  5x⁴  2x³  7x²  39 (5x⁴ + 10x³  20x²)  12x³ + 13x²  39 (12x³  24x² + 48x)  37x²  48x  39 (37x² + 74x  148)  122x + 109

The remainder:
 The degree of the remaining polynomial (122x + 109) is less than the degree of the divisor (x²+2x4), 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)