math:polynomial_division

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====== Polynomial Division ======
===== Usage =====
  - Solving polynomials that can't be grouped
  - Check if roots are valid (with Remainder Theorem)
  - Factoring polynomials to be able to solve as quadratics with quadratic formula, factoring, or completing the square.

===== Universal truths =====
==== Remainder Theorem ====
When you divide a polynomial by a number, the remainder is the value of the divisor function at that number.
==== Factor Theorem ====
If the remainder is zero, the divisor is a factor of the dividend.

===== Long division method =====
For each term in the dividend, multiply the divisor so that the first term of the divisor equals that of the dividend's. The multiplier is the term in your quotient.


===== Synthetic division method =====
Synthetic division is a shortcut that works only on **linear** divisors. First, solve for x with the divisor, synthetic division uses the zero of the function as the divisor. Then, for the dividend, use the coefficient of each term for every degree. Carry down the leftmost coefficient, that will be the starting value. For each empty quotient value, multiply the one before it with the divisor, and place it underneath the corresponding value for the dividend, add the two together and that will be the next quotient value. Repeat until solved. The rightmost quotient value is the remainder.

Trace:

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