6. If
3
3 2
f x mx gx x
is divided by
x 1
, the remainder is 3. If
f x
is divided by
x 3
, the remainder is – 1. What are the values of m and g?
To find the values of m and g, we can use the property that when a polynomial f(x) is divided by (x-a), the remainder is equal to f(a).
1. Given that f(x) is divided by (x-1) and the remainder is 3, we can write:
f(1) = 3
Substituting 1 into the expression for f(x), we get:
f(1) = m(1) + g(1) - 1
3 = m + g - 1
2. Given that f(x) is divided by (x+3) and the remainder is -1, we can write:
f(-3) = -1
Substituting -3 into the expression for f(x), we get:
f(-3) = m(-3) + g(-3) - (-3)
-1 = -3m - 3g + 3
Now we have a system of two equations:
1. 3 = m + g - 1
2. -1 = -3m - 3g + 3
To solve this system, we can use substitution or elimination method.
Let's use elimination method:
Multiply equation 1 by 3:
3(3) = 3(m + g - 1)
9 = 3m + 3g - 3
Rewrite equation 2:
-1 = -3m - 3g + 3
Add equation 1 and equation 2 together:
9 + (-1) = (3m + 3g - 3) + (-3m - 3g + 3)
8 = 0
This equation is not possible. It means there are no values of m and g that satisfy the given conditions.
To find the values of m and g, we can use the remainder theorem. According to the remainder theorem, if we divide a polynomial f(x) by a linear factor (x - a), the remainder is equal to f(a).
Given that the remainder when f(x) is divided by (x - 1) is 3, we can substitute x = 1 into the equation:
f(1) = m(1) + g(1) - 1 = 3
Simplifying this equation, we get:
m + g - 1 = 3
Similarly, given that the remainder when f(x) is divided by (x + 3) is -1, we can substitute x = -3 into the equation:
f(-3) = m(-3) + g(-3) - (-3) = -1
Simplifying this equation, we get:
-3m - 3g + 3 = -1
Now we have a system of equations:
m + g - 1 = 3
-3m - 3g + 3 = -1
We can solve this system of equations using any method, such as substitution or elimination. Let's use the elimination method.
Multiplying the first equation by 3, we get:
3m + 3g - 3 = 9
Adding this equation to the second equation, we can eliminate the g term:
(3m - 3m) + (3g - 3g) + (-3 + 3) = 9 + (-1)
0 + 0 + 0 = 8
0 = 8
This equation is not true, which means the system of equations is inconsistent. There is no solution for m and g that satisfies both conditions.
Therefore, there are no values of m and g that satisfy the given conditions.
To find the values of m and g, we need to use the remainder theorem. The remainder theorem states that if a polynomial f(x) is divided by (x - a), the remainder is f(a).
Let's first solve for m:
When f(x) is divided by (x - 1), the remainder is 3.
So, we need to find f(1) and equate it to 3.
f(1) = m(1) + g(1) - 1
Since the remainder is 3, we have:
3 = m + g - 1
Next, let's solve for g:
When f(x) is divided by (x + 3), the remainder is -1.
So, we need to find f(-3) and equate it to -1.
f(-3) = m(-3) + g(-3) - (-3)
Since the remainder is -1, we have:
-1 = -3m - 3g + 3
We now have a system of equations:
3 = m + g - 1
-1 = -3m - 3g + 3
To solve this system of equations, we can use substitution or elimination method.
Using the elimination method:
Multiply the first equation by 3 and the second equation by -1:
9 = 3m + 3g - 3
1 = 3m + 3g - 3
Subtract the second equation from the first equation:
9 - 1 = (3m + 3g - 3) - (3m + 3g - 3)
8 = 0
This is not possible. There is no solution that satisfies the given conditions.