Long Division with Linear Factors x - a
If three animated example below is too quick for you to follow, rewrite the calculations in them on a piece of paper.
A Polynomial Division Theorem. If p(x) is a polynomial of degree n > 1 and a is real number, then there exists a polynomial q(x) of degree n-1 and real number r such that p(x) = q(x) (x-a) + r. Moreover r = 0 when and only when p(a) = 0.
This theorem can be proven by the method of mathematical induction, a proof method met in or before calculus. Since that method is not at our disposable, we will give a few examples to indicate how the polynomial quotient polynomial q(x) and the remainder r (a real number or a polynomial of degree 0) can be calculated. That is sufficient in practice.
Example 1 of 3, check included.
In the above check, the appearance of expressions that appeared in the long division. That is due to the order of multiplication. of q(x) and x-2.
In the following example to save space, to use fewer rows, the multiplication was done (added up) in a different order.
Example 2 of 3, check included.
Example 3 of 3, check included.
