ifx^3+mx^2+nx+6 has x-2 as a factor and leaves remainder 3 when divided by x-3, find the values of m and n.
Ajib sa answer h😂
To find the values of m and n, we can use the remainder theorem and the factor theorem.
Given that x-2 is a factor of ifx^3 + mx^2 + nx + 6, we know that if we substitute x = 2 into the polynomial, the result should be zero.
Substituting x = 2 into the polynomial:
(2)^3m + (2)^2m + 2n + 6 = 0
8m + 4m + 2n + 6 = 0
12m + 2n + 6 = 0
Now, we are also given that the polynomial leaves a remainder of 3 when divided by x-3. Using the remainder theorem, we know that if we substitute x = 3 into the polynomial, the result should be equal to the remainder.
Substituting x = 3 into the polynomial:
(3)^3m + (3)^2m + 3n + 6 = 3
27m + 9m + 3n + 6 = 3
36m + 3n = -3
We now have a system of equations:
12m + 2n + 6 = 0 ----(1)
36m + 3n = -3 ----(2)
To solve this system, we can multiply equation (1) by 3:
36m + 6n + 18 = 0 ----(3)
Now we can subtract equation (3) from equation (2) to eliminate the variable m:
(36m + 3n) - (36m + 6n + 18) = -3 - 18
3n - 6n = -21
-3n = -21
n = 7
Substituting the value of n=7 into equation (1):
12m + 2(7) + 6 = 0
12m + 14 + 6 = 0
12m + 20 = 0
12m = -20
m = -20/12
m = -5/3
Therefore, the values of m and n are m = -5/3 and n = 7.
To find the values of m and n, we can use the Remainder Theorem and the Factor Theorem.
Since x-2 is a factor of the polynomial, substituting x=2 into the polynomial should give us a remainder of 0. Let's do that:
ifx^3 + mx^2 + nx + 6 = 0
if(2)^3 + m(2)^2 + n(2) + 6 = 0
8 + 4m + 2n + 6 = 0
4m + 2n + 14 = 0
2m + n + 7 = 0 (Equation 1)
Now, we are given that the polynomial leaves a remainder of 3 when divided by x-3. This means that when we substitute x=3 into the polynomial, we should get a remainder of 3.
ifx^3 + mx^2 + nx + 6 = 3
if(3)^3 + m(3)^2 + n(3) + 6 = 3
27 + 9m + 3n + 6 = 3
9m + 3n + 33 = 3
9m + 3n + 30 = 0 (Equation 2)
Now we have two equations (Equation 1 and Equation 2) with two variables (m and n). We can solve these equations simultaneously to find the values of m and n.
Subtracting Equation 1 from Equation 2:
(9m + 3n + 30) - (2m + n + 7) = 0
9m + 3n - 2m - n + 30 - 7 = 0
7m + 2n + 23 = 0
Now, rearranging the equation:
7m + 2n = -23 (Equation 3)
We now have a system of linear equations with Equation 1 and Equation 3. We can solve this system to find the values of m and n.
Multiplying Equation 1 by 2 and subtracting it from Equation 3:
(7m + 2n) - 2(2m + n) = -23
7m + 2n - 4m - 2n = -23
3m = -23
Dividing both sides by 3:
m = -23/3
Substituting the value of m into Equation 1 to find n:
2(-23/3) + n + 7 = 0
-46/3 + n + 7 = 0
n = -7 + 46/3
n = -7 + 46/3
n = -7 + (46/3)
n = -7 + 46/3
Therefore, the values of m and n are:
m = -23/3
n = -7 + 46/3
let f(x) = x^3 + mx^2 + nx + 6
given:
f(2) = 0
= 8 + 4m + 2n + 6 = 0 ---> 2m + n = -7
given: f(3) = 3
= 27 + 9m + 3n + 6 = 3 --- 3m + n = -10
subtract the two equations to get : m = -3
sub that into 2m+n=-7
-6 + n = -7
n = -1
m= -3, n = -1