Polynomial long division

In algebra, synthetic division is an algorithm similar to long division for dividing a polynomial into another polynomial of a larger degree. It can be done easily by hand, because it works by seperating an otherwise complex division problem into smaller problems.

For any polynomials f(x) and g(x), g(x) being of lesser degree than f(x), there exist unique polynomials q(x) and r(x) such that

$f(x)=g(x)*q(x)+r(x)$

If both sides are divided by g(x), then it becomes:

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

Synthetic division will find the quotient (q(x)) and remainder (r(x)) given a dividend (f(x)) and divisor (g(x)). The problem is written down like this:

$g(x){\overline {)f(x)}}$

When the problem is written, all the terms with exponents less than the largest one must be written, even if their coefficients are zero.

Example

Find:

${\frac {x^{3}-12x^{2}-42}{x-3}}$

The problem is written like this (note that the x term is included):

$x-3{\overline {)x^{3}-12x^{2}+0x-42}}$

1. Divide the first term of the dividend by the first term of the divisor. Place the result above the bar (x³ ÷ x = x²).

${\begin{matrix}x^{2}\\\qquad \qquad \quad x-3{\overline {)x^{3}-12x^{2}+0x-42}}\end{matrix}}$

2. Multiply the divisor by the term you just wrote. Write the result under the first two terms of the dividend (x² * (x-3) = x³ - 3x²).

${\begin{matrix}x^{2}\\\qquad \qquad \quad x-3{\overline {)x^{3}-12x^{2}+0x-42}}\\\qquad \;\;x^{3}-3x^{2}\end{matrix}}$

3. Subtract the second term of the result you just got from the second term of the dividend and write the result under both of them. This can be tricky at times, because of the sign. (-12x² - (-3x²) = -12x² + 3x² = -9x²) Then, "pull down" the next term from the dividend.

${\begin{matrix}x^{2}\\\qquad \qquad \quad x-3{\overline {)x^{3}-12x^{2}+0x-42}}\\\qquad \;\;{\underline {x^{3}-\quad 3x^{2}}}\\\qquad \qquad \qquad \quad \;-9x^{2}+0x\end{matrix}}$

4. Repeat the last three steps, except this time use the two terms that you have just written as the dividend.

${\begin{matrix}\;x^{2}-9x\\\qquad \quad x-3{\overline {)x^{3}-12x^{2}+0x-42}}\\\;\;{\underline {\;\;x^{3}-\;\;3x^{2}}}\\\qquad \qquad \quad \;-9x^{2}+0x\\\qquad \qquad \quad \;{\underline {-9x^{2}+27x}}\\\qquad \qquad \qquad \qquad \qquad -27x-42\end{matrix}}$

5. Repeat step 4. This time, there is nothing to pull down.

${\begin{matrix}\qquad \quad \;\,x^{2}\;-9x\quad -27\\\qquad \quad x-3{\overline {)x^{3}-12x^{2}+0x-42}}\\\;\;{\underline {\;\;x^{3}-\;\;3x^{2}}}\\\qquad \qquad \quad \;-9x^{2}+0x\\\qquad \qquad \quad \;{\underline {-9x^{2}+27x}}\\\qquad \qquad \qquad \qquad \qquad -27x-42\\\qquad \qquad \qquad \qquad \qquad {\underline {-27x+81}}\\\qquad \qquad \qquad \qquad \qquad \qquad \;\;-123\end{matrix}}$

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

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