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This essential trains for: AMC-10, AMC-12, AIME, SAT-II, Math Kangaroo 9-10.

A degree n polynomial Pn(x) is divisible by the degree m polynomial Dn(x) with n ≥ m and Dm(x) ≠ 0, if and only if there exists a degree n-m polynomial Qn-m(x) such that:

equation

Remainder Theorem If a degree n polynomial Pn(x) is divided by the degree 1 monic polynomial x-a, then the remainder from the division is Pn(a).

From the division theorem, we find that the degree of the remainder must be smaller than the degree of the divisor. Since the divisor has degree 1 then the remainder must have degree 1, i.e. it must be a constant.

We can now write the division theorem as:

equation

Substitute a for x:

equation

where from clearly:

equation

Divisibility criterion from the remainder theorem it is obvious that, if:

equation

then Pn(x) is divisible by the degree 1 monic polynomial x-a.

Notation: D divides P (or P is divisible by D)

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Properties of divisibility