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This essential trains for: AMC-10, AMC-12, AIME.

Some simple facts:

The sum of the coefficients of a polynomial is, obviously: whereas the sum with alternating signs is: For a polynomial to have a root of zero, x must be a factor and, therefore, the free term must be zero: For zero to be a double root, the coefficient of x must also be zero.

For a polynomial to be divisible by a linear factor, the remainder theorem (Bezout) is what you need to apply.

For other cases of divisibility some less known facts may help:

For a polynomial to be divisible by the quadratic factor: the complex-imaginary roots of unity ω must be roots: Euclid's algorithm can be applied to find the GCD of two polynomials.

Take two polynomials that satisfy the conditions:   Apply the algorithm:

Step 1: Divide the higher degree polynomial by the lower degree polynomial.

Step 2: If the remainder is zero, then the GCD is the divisor. End.

Step 3: If the remainder is not zero, then set the divisor to be dividend and the remainder to be divisor. Go to step 1.

At the end of the algorithm, the last non-zero remainder is the greatest common divisor of the two polynomials.

If GCD = 1 then the polynomials are coprime.

Example of polynomial analysis:

A polynomial P is divisible by x-2 and P(7)=9. What is the remainder when we divide the polynomial by: Solution: By the division theorem, if the divisor is quadratic, the remainder must be linear. Assume the remainder to have the form: We notice that: and we use the division theorem: Since:  we can use this linear 2x2 system to determine the coefficients of the remainder:   and the remainder is the linear polynomial: 