Determine the values of m and n for 𝑓(𝑥) = 𝑚𝑥3 + 12𝑥2 + 𝑛𝑥 − 3 given that the remainder when dividing by (𝑥 + 3) is zero, and when divided by (𝑥 − 2) the remainder is 85.

Solution: By Remainder Theorem, which states that if a polynomial f(x) is divided by (x-a), the remainder is f(a). From the first condition, m\left(-3\right)^3+12\left(-3\right)^2+n\left(-3\right)-3=0 -27m+108-3n-3=0 -27m+105-3n=0 27m+3n=105 From the second condition, m\left(2\right)^3+12\left(2\right)^2+n\left(2\right)-3=85 8m+48+2n-3=85 8m+45+2n=85 8m+2n=40 The problem now is a system of linear equations. Subtracting three times equation 2 to two times equation 1, 2\left(27m+3n\right)-3\left(8m+2n\right)=2\left(105\right)-3\left(40\right) 54m+6n-24m-6n=210-120 30m=90 m=3 Substituting m to either equation (in this solution, using equation 2), 8\cdot3+2n=40 24+2n=40 2n=16 n=8 Answers: m=3 n=8