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Polynomial long division

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In algebra, polynomial long division is an algorithm for dividing a polynomial by another polynomial of the same or lower degree, a generalised version of the familiar arithmetic technique called long division. It can be done easily by hand, because it separates an otherwise complex division problem into smaller ones.

For any polynomials f(x) and g(x), with g(x) not identical to zero, there exist unique polynomials q(x) and r(x) such that

f ( x ) g ( x ) = q ( x ) + r ( x ) g ( x ) {\displaystyle {\frac {f(x)}{g(x)}}=q(x)+{\frac {r(x)}{g(x)}}}

with r(x) having smaller degree than g(x).

Synthetic division will find the quotient q(x) and remainder r(x) given a numerator f(x) and nonzero denominator g(x). The problem is written down like a regular (non-algebraic) long division problem:

g ( x ) | f ( x ) ¯ {\displaystyle g(x){\overline {\vert f(x)}}} .

All terms with exponents less than the largest one must be written out explicitly, even if their coefficients are zero.

Example

Find:

x 3 − 12 x 2 − 42 x − 3 {\displaystyle {\frac {x^{3}-12x^{2}-42}{x-3}}}

The problem is written like this (note that, as explained above, the x term is included explicitly, regardless of the coefficient):

x − 3 | x 3 − 12 x 2 + 0 x − 42 ¯ {\displaystyle x-3{\overline {\vert x^{3}-12x^{2}+0x-42}}}

The quotient and remainder can then be determined as follows:

  1. Divide the first term of the numerator by the highest term of the denominator. Place the result above the bar (x ÷ x = x· x = x = x).
    x 2 x − 3 | x 3 − 12 x 2 + 0 x − 42 ¯ {\displaystyle {\begin{matrix}x^{2}\\\qquad \qquad \quad x-3{\overline {\vert x^{3}-12x^{2}+0x-42}}\end{matrix}}}
  2. Multiply the denominator by the result you just obtained (the first term of the eventual quotient). Write the result under the first two terms of the numerator (x · (x − 3) = x − 3x).
    x 2 x − 3 | x 3 − 12 x 2 + 0 x − 42 ¯ x 3 − 3 x 2 {\displaystyle {\begin{matrix}x^{2}\\\qquad \qquad \quad x-3{\overline {\vert x^{3}-12x^{2}+0x-42}}\\\qquad \;\;x^{3}-3x^{2}\end{matrix}}}
  3. Subtract the product you just obtained from the appropriate terms of the original numerator, and write the result underneath. This can be tricky at times, because of the sign. ((x − 12x) − (x − 3x) = −12x + 3x = −9x) Then, "bring down" the next term from the numerator.
    x 2 x − 3 | x 3 − 12 x 2 + 0 x − 42 ¯ x 3 − 3 x 2 _ − 9 x 2 + 0 x {\displaystyle {\begin{matrix}x^{2}\\\qquad \qquad \quad x-3{\overline {\vert x^{3}-12x^{2}+0x-42}}\\\qquad \;\;{\underline {x^{3}-3x^{2}}}\\\qquad \qquad \qquad \quad \;-9x^{2}+0x\end{matrix}}}
  4. Repeat the previous three steps, except this time use the two terms that you have just written as the numerator.
    x 2 − 9 x x − 3 | x 3 − 12 x 2 + 0 x − 42 ¯ x 3 − 3 x 2 _ − 9 x 2 + 0 x − 9 x 2 + 27 x _ − 27 x − 42 {\displaystyle {\begin{matrix}\;x^{2}-9x\\\qquad \quad x-3{\overline {\vert x^{3}-12x^{2}+0x-42}}\\\;\;{\underline {\;\;x^{3}-\;\;3x^{2}}}\\\qquad \qquad \quad \;-9x^{2}+0x\\\qquad \qquad \quad \;{\underline {-9x^{2}+27x}}\\\qquad \qquad \qquad \qquad \qquad -27x-42\end{matrix}}}
  5. Repeat step 4. This time, there is nothing to "pull down".
    x 2 − 9 x − 27 x − 3 | x 3 − 12 x 2 + 0 x − 42 ¯ x 3 − 3 x 2 _ − 9 x 2 + 0 x − 9 x 2 + 27 x _ − 27 x − 42 − 27 x + 81 _ − 123 {\displaystyle {\begin{matrix}\qquad \quad \;\,x^{2}\;-9x\quad -27\\\qquad \quad x-3{\overline {\vert x^{3}-12x^{2}+0x-42}}\\\;\;{\underline {\;\;x^{3}-\;\;3x^{2}}}\\\qquad \qquad \quad \;-9x^{2}+0x\\\qquad \qquad \quad \;{\underline {-9x^{2}+27x}}\\\qquad \qquad \qquad \qquad \qquad -27x-42\\\qquad \qquad \qquad \qquad \qquad {\underline {-27x+81}}\\\qquad \qquad \qquad \qquad \qquad \qquad \;\;-123\end{matrix}}}

The polynomial above the bar is the quotient, and the number left over (−123) is the remainder.

x 3 − 12 x 2 − 42 x − 3 = x 2 − 9 x − 27 − 123 x − 3 {\displaystyle {\frac {x^{3}-12x^{2}-42}{x-3}}=x^{2}-9x-27-{\frac {123}{x-3}}}

The long division algorithm learned in elementary arithmetic classes can be viewed as a special case of the above algorithm.

Synthetic division

Main article: Ruffini's rule

Synthetic division is a method of performing polynomial long division without having to maintain long records of the process of long division as above, though the processes are still the same. It, however, only deals with division by monic linear polynomials -- that is, binomials of the form (x-b), where b is any rational number.

The change of sign of b when dividing by (x + b) (i.e. −b is written to the left of the bar rather than b) allows us to do additions rather than subtractions that are found in regular long division. This reduces the chance for error when the division is done by hand.

Synthetic division is also called division through Ruffini's rule and was described by Paolo Ruffini in 1809.

Performing the same example as before:

x − 3 | x 3 − 12 x 2 + 0 x − 42 ¯ {\displaystyle x-3{\overline {\vert x^{3}-12x^{2}+0x-42}}}

We concern ourselves basically with the coefficients. We write

3 | 1 − 12 0 − 42 {\displaystyle {\begin{matrix}3&|&1&-12&0&-42\end{matrix}}}

Note the change of sign from −3 to 3, the sign doesn't actually change, the value you are using (3 in this case) is the value that x would have to be in order for the polynomial to equal 0. "Drop" the first coefficient after the bar to the third row.

3 | 1 − 12 0 − 42 | | 1 {\displaystyle {\begin{matrix}3&|&1&-12&0&-42\\&|&&&&\\&|&1&&&\\\end{matrix}}}

Multiply the dropped number by the number before the bar, and place it in the next column.

3 | 1 − 12 0 − 42 | 3 | 1 {\displaystyle {\begin{matrix}3&|&1&-12&0&-42\\&|&&3&&\\&|&1&&&\\\end{matrix}}}

Perform an addition in that column.

3 | 1 − 12 0 − 42 | 3 | 1 − 9 {\displaystyle {\begin{matrix}3&|&1&-12&0&-42\\&|&&3&&\\&|&1&-9&&\\\end{matrix}}}

Repeat the previous two steps, the following is obtained

3 | 1 − 12 0 − 42 | 3 − 27 − 81 | 1 − 9 − 27 − 123 {\displaystyle {\begin{matrix}3&|&1&-12&0&-42\\&|&&3&-27&-81\\&|&1&-9&-27&-123\\\end{matrix}}}

All the numbers on the last row besides the farthest right correspond to coefficients in the quotient; the last number indicates a remainder. The terms are written with increasing degree from right to left, starting to the left of the remainder with degree 0.

The result of our division is:

x 3 − 12 x 2 − 42 x − 3 = x 2 − 9 x − 27 − 123 x − 3 {\displaystyle {\frac {x^{3}-12x^{2}-42}{x-3}}=x^{2}-9x-27-{\frac {123}{x-3}}}

Higher degree synthetic division

The technique of synthetic division described above works only with linear denominators. A similar shortcut method exists for dividing by a quadratic or higher degree monic polynomial.

For instance, let's try to perform the following division:

x 3 − 12 x 2 − 42 x 2 + x − 3 {\displaystyle {\frac {x^{3}-12x^{2}-42}{x^{2}+x-3}}}

Create the table as with ordinary synthetic division, using the negative coefficients of the denominator following the leading term. We write

− 1 3 | 1 − 12 0 − 42 {\displaystyle {\begin{matrix}-1&3&|&1&-12&0&-42\end{matrix}}}

Next, underline the leading coefficient of the right-hand side, multiply it by the left-hand coefficients and write the products beneath the next columns on the right.

− 1 3 | 1 _ − 12 0 − 42 | − 1 3 {\displaystyle {\begin{matrix}-1&3&|&{\underline {1}}&-12&0&-42\\&&|&&-1&3&\\\end{matrix}}}

Now perform an addition:

− 1 3 | 1 _ − 12 0 − 42 | − 1 3 | − 13 3 − 42 {\displaystyle {\begin{matrix}-1&3&|&{\underline {1}}&-12&0&-42\\&&|&&-1&3&\\&&|&&-13&3&-42\\\end{matrix}}}

Repeat the preceding two steps upon the bottom row of the right-hand side.

− 1 3 | 1 _ − 12 0 − 42 | − 1 3 | − 13 _ 3 − 42 | 13 − 39 | 16 − 81 {\displaystyle {\begin{matrix}-1&3&|&{\underline {1}}&-12&0&-42\\&&|&&-1&3&\\&&|&&{\underline {-13}}&3&-42\\&&|&&&13&-39\\&&|&&&16&-81\\\end{matrix}}}

The underlined numbers correspond to the coefficients of the quotient, the remaining numbers in the bottom row correspond to the coefficients of the remainder. The terms are written with increasing degree from right to left, with the last terms of the quotient and remainder each having degree 0.

The result of our division is

x 3 − 12 x 2 − 42 x 2 + x − 3 = x − 13 + 16 x − 81 x 2 + x − 3 . {\displaystyle {\frac {x^{3}-12x^{2}-42}{x^{2}+x-3}}=x-13+{\frac {16x-81}{x^{2}+x-3}}.}

See also

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