Polynomial Long Division: (5x⁴  2x³  7x²  39) ÷ (x² + 2x  4)
This article will walk you 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 dividing numbers, we are dividing polynomials. The goal is to find the quotient and remainder of the division.
Steps to Perform Polynomial Long Division

Set up the division. Write the dividend (5x⁴  2x³  7x²  39) inside the division symbol and the divisor (x² + 2x  4) outside.
____________ x² + 2x  4  5x⁴  2x³  7x²  39

Divide the leading terms. Focus on the leading terms of the dividend and the divisor (5x⁴ and x²). Ask yourself: "What do I need to multiply x² by to get 5x⁴?" The answer is 5x². Write this term above the division symbol, aligned with the x⁴ term.
5x² _________ x² + 2x  4  5x⁴  2x³  7x²  39

Multiply the divisor. Multiply the divisor (x² + 2x  4) by the term you just wrote (5x²). This gives you 5x⁴ + 10x³  20x². Write this result below the dividend, aligning terms with corresponding powers.
5x² _________ x² + 2x  4  5x⁴  2x³  7x²  39 5x⁴ + 10x³  20x²

Subtract. Subtract the expression you just wrote from the dividend. Remember to change the signs of the terms you are subtracting.
5x² _________ x² + 2x  4  5x⁴  2x³  7x²  39 5x⁴ + 10x³  20x²  12x³ + 13x²  39

Bring down the next term. Bring down the next term of the dividend (39).
5x² _________ x² + 2x  4  5x⁴  2x³  7x²  39 5x⁴ + 10x³  20x²  12x³ + 13x²  39

Repeat steps 25. Now focus on the new leading terms: 12x³ and x². What do you need to multiply x² by to get 12x³? The answer is 12x. Write 12x above the division symbol, aligned with the x³ term.
5x²  12x _________ x² + 2x  4  5x⁴  2x³  7x²  39 5x⁴ + 10x³  20x²  12x³ + 13x²  39

Continue until the degree of the remainder is less than the degree of the divisor. Repeat steps 25 until the degree of the polynomial you get after subtraction is less than the degree of the divisor (x² + 2x  4).
5x²  12x + 19 _________ x² + 2x  4  5x⁴  2x³  7x²  39 5x⁴ + 10x³  20x²  12x³ + 13x²  39 12x³  24x² + 48x  37x² + 48x  39 37x² + 74x  148  26x + 109
Result
The result of the division is:
Quotient: 5x²  12x + 19
Remainder: 26x + 109
Therefore, we can write:
(5x⁴  2x³  7x²  39) ÷ (x² + 2x  4) = 5x²  12x + 19 + (26x + 109) / (x² + 2x  4)