Answer:
m = 9 and n = 3
Explanation:
To find the values of m and n in the polynomial 2x³ + mx² + nx - 14 such that (x - 1) and (x + 2) are factors, we can use the factor theorem.
According to the factor theorem, if (x - r) is a factor of a polynomial, then the polynomial will be equal to 0 when we substitute x = r.
Using this theorem, we can find the values of m and n by substituting x = 1 and x = -2 into the given polynomial and setting them equal to zero.
For (x - 1) = 0,
we have:
2(1)³ + m(1)² + n(1) - 14 = 0
2 + m + n - 14 = 0
m + n = 12 -- (Equation 1)
For (x + 2) = 0,
we have:
2(-2)³ + m(-2)² + n(-2) - 14 = 0
-16 + 4m - 2n - 14 = 0
4m - 2n = 30 -- (Equation 2)
Now we have a system of equations (Equation 1 and Equation 2) to solve simultaneously.
From Equation 1, we can express m in terms of n:
m = 12 - n
Substituting this into Equation 2:
4(12 - n) - 2n = 30
48 - 4n - 2n = 30
48 - 6n = 30
-6n = 30 - 48
-6n = -18
n = -18 / -6
n = 3
Substituting n = 3 into Equation 1:
m + 3 = 12
m = 12 - 3
m = 9
Therefore, the values of m and n that satisfy the given conditions are m = 9 and n = 3.