Polynomial long division is a method for dividing one polynomial by another, especially when the divisor has more than one term. It works much like numerical long division because you divide the leading term, multiply back, subtract, and bring down the next term. This skill matters because it helps simplify rational expressions, factor polynomials, and solve polynomial equations.
It also shows how a dividend can be rewritten as a quotient plus a remainder part.
Key Facts
- Dividend = Divisor × Quotient + Remainder
- If P(x) is divided by D(x), then P(x)/D(x) = Q(x) + R(x)/D(x)
- At each step, divide the leading term of the current polynomial by the leading term of the divisor.
- Multiply the entire divisor by the new quotient term, then subtract the result from the current polynomial.
- The division stops when the degree of the remainder is less than the degree of the divisor.
- Example result: (2x^3 + 3x^2 - 11x - 6)/(x + 3) = 2x^2 - 3x - 2
Vocabulary
- Dividend
- The polynomial being divided in a division problem.
- Divisor
- The polynomial you divide by.
- Quotient
- The polynomial result produced by the division process before including any remainder term.
- Remainder
- The polynomial left over when the division cannot continue because its degree is smaller than the divisor's degree.
- Degree
- The highest exponent of the variable in a polynomial with a nonzero coefficient.
Common Mistakes to Avoid
- Forgetting placeholder terms, such as writing x^3 + 5 instead of x^3 + 0x^2 + 0x + 5, makes columns misalign and leads to wrong subtraction.
- Dividing by the wrong term is incorrect because each new quotient term must come from the leading term of the current polynomial divided by the leading term of the divisor.
- Subtracting without changing every sign causes errors because the entire product of divisor and quotient term must be subtracted.
- Stopping too early is wrong if the current remainder still has degree greater than or equal to the degree of the divisor, because another division step is still possible.
Practice Questions
- 1 Divide 3x^3 + 5x^2 - 4x + 8 by x + 2. Find the quotient and remainder.
- 2 Divide 2x^4 - x^3 + 0x^2 + 7x - 10 by x^2 - 1. Find the quotient and remainder.
- 3 A student says that the remainder when dividing by x^2 + 3 must always be a constant. Explain whether this is correct and justify your answer using degree.