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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:

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:

Substitute a for x:

where from clearly:

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

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

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

Properties of divisibility

• Transitivity

• Reflexivity

• If D|P and P|D then there is a constant complex value g ≠ 0 such that:

• Any non-zero complex constant divides any polynomial
• For any non-zero complex constant g

• If a polynomial divides two other polynomials then it divides their sum: