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

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

85 = m(2)^3 + 12(2)^2 + n(2) -3

85 = 8m +48 +2n -3

85-48+3 = 8m+2n

40/2 = 8m +2n /2

20 = 4m + n (Equation 1)

0 = m(-3)^3 +12(-3)^2 +n(-3) -3

0 = -27m +108 -3n -3

0 = -27m +105 -3n

-105/-3 = -27m -3n /-3

35 = 9m + n (equation 2)

(equation 1 – equation 2) 20 = 4m + n – 35 = 9m + n —> -15 = 5m divide by 5 and you get m = -3

now plug m into equation 1 to get n

20 = 4(-3) + n —> n= 32