Math · College algebra · Concept
Polynomial division and the remainder theorem
Dividing a polynomial P(x) by a divisor D(x) gives a quotient Q(x) and a remainder R(x) of lower degree than D, with P(x) = D(x)Q(x) + R(x). Long division works for any divisor; synthetic division is a shortcut for divisors of the form x − c. The remainder theorem says that the remainder on dividing by x − c is P(c), so x − c is a factor exactly when P(c) = 0: the factor theorem.
The division statement
Just as 17 = 5 × 3 + 2, polynomial division writes P(x) = D(x)Q(x) + R(x), where the remainder has lower degree than the divisor. Dividing both sides by D(x) gives P/D = Q + R/D.
Long division
Divide the leading term of what remains by the leading term of the divisor, multiply the divisor by the result, and subtract. Repeat until what remains has lower degree than the divisor. Write 0 for any missing power so that the columns line up.
Synthetic division
For a divisor x − c, write c and the coefficients of P. Bring down the first coefficient; then repeatedly multiply by c, add the product to the next coefficient and write the sum. The last sum is the remainder, and the others are the coefficients of the quotient, one degree lower than P.
The remainder theorem
Setting x = c in P(x) = (x − c)Q(x) + R makes the first term vanish, so the remainder is R = P(c). When only the remainder is needed, evaluating P(c) is often quicker than dividing.
The factor theorem
x − c is a factor of P(x) exactly when P(c) = 0. Once one zero is known, dividing it out leaves a polynomial of lower degree, which may be a quadratic that factoring or the quadratic formula finishes.
Common mistakes
- Skipping a missing power in long or synthetic division: write 0 in its place.
- Using −c instead of c in synthetic division: to divide by x + 2, use c = −2.
- Stopping long division too early: continue until the remainder has lower degree than the divisor.
- Leaving out the remainder when writing the result as a single expression: it belongs over the divisor, as R/D.
Key terms
- Polynomial long division
- Dividing one polynomial by another with divide, multiply, subtract and bring-down steps, so that dividend = divisor × quotient + remainder. The remainder is 0 or has a lower degree than the divisor.
- Synthetic division
- A shortcut for dividing a polynomial by x − c using only its coefficients. Write a 0 for every missing power, or the columns won’t line up.
- Remainder theorem
- When a polynomial p(x) is divided by x − c, the remainder is p(c). So substituting c gives the same number as dividing.
- Factor theorem
- x − c is a factor of a polynomial p exactly when p(c) = 0. A graph that only looks as if it crosses at c isn’t proof.
- Polynomial
- An expression made by adding terms, each a number times whole-number powers of the variables, such as 3x² − 5x + 2. An expression with a variable in a denominator, under a root or inside a function like sin is not a polynomial.
- Root of an equation
- A value that makes an equation true. A root of f(x) = 0 is also called a zero of f, and it must be in the domain.
Work through an example
Divide 2x³ − 3x² − 11x + 6 by x − 3 using synthetic division, then find all the zeros of the polynomial.
Divide polynomials with synthetic division →Sources and scope
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Open the example inputs, change a value and keep a useful result on your board.
Graph the cubic Check the factors in Math Open worked example on a board Zeros, factors and remainders in Math ReferenceYour existing work stays on this device. Examples open as editable copies.