The user-friendly version of this content is available here.

The following content is copyright (c) 2009-2013 by Goods of the Mind, LLC.

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

Some simple facts:

The sum of the coefficients of a polynomial is, obviously:

equation

whereas the sum with alternating signs is:

equation

For a polynomial to have a root of zero, x must be a factor and, therefore, the free term must be zero:

equation

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:

equation

the complex-imaginary roots of unity ω must be roots:

equation

Euclid's algorithm can be applied to find the GCD of two polynomials.

Take two polynomials that satisfy the conditions:

equation

equation

equation

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:

equation

Solution: By the division theorem, if the divisor is quadratic, the remainder must be linear. Assume the remainder to have the form:

equation

We notice that:

equation

and we use the division theorem:

equation

Since:

equation

equation

we can use this linear 2x2 system to determine the coefficients of the remainder:

equation

equation

equation

and the remainder is the linear polynomial:

equation