Question
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.
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Answer to a math question 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.
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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
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