JentooPrecalculus

AP® Precalculus · Unit 1 · Topic 1.11

Polynomial Long Division

Polynomial long division splits one polynomial into a quotient and remainder using the same subtract-and-bring-down process as number division, and the remainder it leaves equals the value of the polynomial at the root of the divisor.

Watch it drawn3 questionsPractice

Dividing polynomials works like dividing numbers because both are built on the same idea: pull out as much of the divisor as fits, subtract, and repeat with what's left. When you divide 157 by 12, you ask how many times 12 fits into the leading chunk, subtract, bring down the next digit, and keep going until the leftover is smaller than the divisor. Polynomial long division does the same thing with terms instead of digits. You match the leading term of the dividend to the leading term of the divisor, multiply the divisor by that matching factor, subtract the result, and bring down the next term. You stop when the degree of what's left is smaller than the degree of the divisor. That leftover piece is the remainder.

Take x3 - 2x + 5 divided by x - 2. Divide x3 by x to get x2, multiply (x-2) by x2 to get x3 - 2x2, and subtract: that leaves 2x2 - 2x + 5. Divide 2x2 by x to get 2x, multiply and subtract to leave 2x + 5. Divide 2x by x to get 2, multiply (x-2) by 2 to get 2x - 4, and subtract to leave a remainder of 9. So x3 - 2x + 5 = (x-2)(x2 + 2x + 2) + 9.

This connects directly to evaluating a polynomial. The Remainder Theorem says that dividing f(x) by (x-k) always leaves a remainder equal to f(k). Check it above: plugging x = 2 into x3 - 2x + 5 gives 8 - 4 + 5 = 9, matching the remainder exactly. This is why finding zeros, factoring, and long division all sit next to each other in this unit: a remainder of 0 means (x-k) divides evenly, which means k is a root of the polynomial.

Key terms

Quotient
The polynomial result of the division, not counting whatever is left over.
Remainder
The leftover polynomial, of lower degree than the divisor, after division is complete.
Remainder Theorem
The rule that dividing f(x) by (x-k) leaves a remainder equal to f(k).
Synthetic division
A shortcut version of long division by (x-k) that uses only the coefficients.

Watch it explained

  1. How it worksWalk me through dividing x³ - 2x + 5 by x - 2 one step at a time.
  2. Worked exampleFind the quotient and remainder when 2x³ + 3x² - 1 is divided by x + 2.
  3. Ties togetherHow does the remainder from long division connect to evaluating the polynomial at that input?

Check yourself

Your turn1 of 2
Find the quotient when f(x) = 4x3 + 6x2 + 9 is divided by (x - 2), as a function of x (ignore the remainder).

End of topic.

Next: The Binomial Theorem →