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This problem trains for: AMC-10, AMC-12, Math Kangaroo 9-10, AIME.

When the polynomial:

equation

is divided by the polynomial of degree 5:

equation

the remainder is of the form:

equation

where the coefficients ak can have any real values.

What is the value of:

equation

The polynomial is equal to the divisor times the quotient plus the remainder:

equation

Since polynomials are defined for all real numbers, we can substitute -1 for x to get:

equation

Since:

We have that:

equation

it means that:

equation

The answer is:

equation