Polynomial Long Division
In short: Polynomial long division divides one polynomial by another using the same divide, multiply, subtract and bring-down loop as long division with whole numbers.
Algebra 2AlgebraA-APR.D.6
What is Polynomial Long Division?
Polynomial long division splits a polynomial \(p(x)\) by a divisor \(d(x)\) to produce a quotient \(q(x)\) and a remainder \(r(x)\), so that \(p(x) = d(x)\,q(x) + r(x)\), with the remainder of lower degree than the divisor.
It works exactly like dividing 2,583 by 7, except each step handles a power of \(x\). It is the method of choice when the divisor has degree 2 or more, or does not have the form \(x – c\), and it proves useful for rewriting rational functions and finding hidden factors.
How to spot it
- A problem asks you to divide a polynomial by a binomial or trinomial and state a quotient and remainder.
- A rational expression has a numerator of higher or equal degree than its denominator and must be rewritten.
- You are testing whether a given expression is a factor of a polynomial.
- The divisor is something like \(x^2 + 1\) or \(2x – 5\), where a shortcut table would not fit.
The method
- Write both polynomials in descending order, adding zero terms for any missing power.
- Divide the leading term of the current dividend by the leading term of the divisor; write the result above.
- Multiply the new quotient term by the whole divisor and subtract, being careful with signs, then bring down the next term.
- Repeat until the remainder has lower degree than the divisor, and check: divisor times quotient plus remainder.
Polynomial Long Division: solved example
Problem
Divide \(2x^3 + 5x^2 – 4x + 7\) by \(x + 3\).
Solution
Step 1: \(2x^3 \div x = 2x^2\). Multiply: \(2x^2(x + 3) = 2x^3 + 6x^2\). Subtract: \(-x^2 – 4x\) after bringing down.
Step 2: \(-x^2 \div x = -x\). Multiply: \(-x(x + 3) = -x^2 – 3x\). Subtract: \(-x + 7\). Step 3: \(-x \div x = -1\). Multiply: \(-1(x + 3) = -x – 3\). Subtract: \(7 – (-3) = 10\).
Quotient \(2x^2 – x – 1\), remainder 10. Check: \((x + 3)(2x^2 – x – 1) + 10 = 2x^3 + 5x^2 – 4x – 3 + 10\).
Try it
Question 1
Divide \(x^2 + 7x + 12\) by \(x + 4\).
Show the answer
The quotient is \(x + 3\) with remainder 0, since \((x + 4)(x + 3) = x^2 + 7x + 12\).
Question 2
Divide \(3x^2 – 2x + 5\) by \(x – 2\).
Show the answer
The quotient is \(3x + 4\) with remainder 13. Check: \((x – 2)(3x + 4) + 13 = 3x^2 – 2x – 8 + 13\).
Polynomial Long Division vs. Synthetic Division
Synthetic division is a compact shortcut that records only coefficients and works only when the divisor looks like \(x – c\). Polynomial long division is the full method and handles any divisor, such as \(x^2 + 1\) or \(2x – 5\). When both apply, they give the same quotient and remainder.
Keep going
Lessons
Worksheets and quizzes
- Dividing Polynomials and the Remainder Theorem – Worksheet
- Dividing Polynomials and the Remainder Theorem – Quiz
Related words
Whole dictionary: A–Z · Algebra 2 · Algebra