polynomial long division

Polynomial long division divides one polynomial by another using the very same staircase procedure you learned for dividing whole numbers — just with x's along for the ride. It is how you split a big polynomial into a quotient plus a remainder.

The cycle repeats: divide the leading term of the current dividend by the leading term of the divisor to get the next quotient term, multiply that term back through the whole divisor, subtract to clear the leading term, bring down the next term, and go again until the remainder has lower degree than the divisor.

The result reads dividend = divisor × quotient + remainder, exactly mirroring 17 = 5 × 3 + 2 with numbers. Two habits prevent grief: write both polynomials in descending order first, and insert a 0 placeholder for any missing power (for instance write x^3 + 0x^2 + 0x - 8) so columns stay aligned.