Given two remainders and their corresponding divisors as well as a factor of a polynomial, the solution to find the polynomial is presented. The polynomial is first written as
P(x) = x^3 + b x^2 + cx + d
since it is a polynomial of degree 3 and its leading coefficient is equal to 1.
The coefficients b, c, and d are found by writing 2 equations using the remainder theorem which states that the remainder of the division P(x) / (x - k) is equal to P(k). Also since (x - 1) is a factor,
P(1) = 0.
We, therefore, end up with a system of three linear equations in three variables b, c, and d which is solved, and hence the polynomial is completely determined once a, b and c are found.
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