How do you divide polynomials with long division?

Before you do any math, you need to set up the problem correctly. Write both polynomials in standard form, which means the terms go from the highest exponent down to the lowest. If the dividend is missing any powers of x, you must add a placeholder with a coefficient of zero. For example, write x^3 - 4 as x^3 + 0x^2 + 0x - 4. This keeps all your terms lined up in the right columns, just like keeping the hundreds, tens, and units places lined up in regular division.

Polynomial long division follows a repeating four-step cycle: Divide, Multiply, Subtract, and Bring down. First, divide the leading term of the dividend by the leading term of the divisor to get the first part of your quotient. Second, multiply that new quotient term by the entire divisor. Third, subtract that result from the dividend. Finally, bring down the next term from the original polynomial and repeat the cycle until the degree of your remainder is less than the degree of your divisor.

The most common mistake happens during the subtraction step. When you subtract the multiplied terms, you are subtracting the entire binomial or polynomial. You have to distribute the negative sign to every single term. A great habit is to write the terms you are subtracting in parentheses with a minus sign outside, then physically change the signs of each term before combining them with the line above. This single habit prevents the vast majority of polynomial division errors.