when the expression x^3+mx^2+nx-5 is divided by x+2 and x+3, the remainders are 2 and 13, respectively. find the value of m and n.
tnx..
To find the values of m and n, we can use the Remainder Theorem. According to the theorem, if a polynomial f(x) is divided by x - a, the remainder is equal to f(a).
Given that the expression x^3 + mx^2 + nx - 5 is divided by x + 2 and x + 3, the remainders are 2 and 13, respectively. We can set up two equations using the Remainder Theorem:
Equation 1: (Substituting x = -2)
(-2)^3 + m(-2)^2 + n(-2) - 5 = 2
-8 + 4m - 2n - 5 = 2
4m - 2n = 15 (Equation 1)
Equation 2: (Substituting x = -3)
(-3)^3 + m(-3)^2 + n(-3) - 5 = 13
-27 + 9m - 3n - 5 = 13
9m - 3n = 45 (Equation 2)
Now, we have a system of two equations (Equation 1 and Equation 2) with two variables (m and n). We can solve this system of equations to find the values of m and n.
Multiplying Equation 1 by 3 and Equation 2 by 2 to eliminate the coefficient of n, we get:
12m - 6n = 45 (Equation 3)
18m - 6n = 90 (Equation 4)
Subtracting Equation 3 from Equation 4, we have:
(18m - 6n) - (12m - 6n) = 90 - 45
6m = 45
m = 7.5
Substituting the value of m in Equation 1, we can solve for n:
4(7.5) - 2n = 15
30 - 2n = 15
-2n = 15 - 30
-2n = -15
n = 7.5
Therefore, the values of m and n are 7.5.
follow the same steps I just showed you in your previous post.
Let me know what you got