5. 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 2
, the remainder is – 7. What are the values of m and g?
To find the values of m and g, we can use the Remainder Theorem. According to the Remainder Theorem, if a polynomial f(x) is divided by (x - c), the remainder is equal to f(c).
Given that f(x) is divided by (x + 1) and the remainder is 3, we have:
f(-1) = 3
Substituting -1 into f(x), we get:
3 = m(-1)^3 + g(-1)^2 - (-1) + 1
3 = -m + g + 2
Simplifying the equation, we get:
-m + g = 1 .........(1)
Now, given that f(x) is divided by (x + 2) and the remainder is -7, we have:
f(-2) = -7
Substituting -2 into f(x), we get:
-7 = m(-2)^3 + g(-2)^2 - (-2) + 2
-7 = -8m + 4g + 2
Simplifying the equation, we get:
-8m + 4g = -9 .........(2)
To find the values of m and g, we need to solve these two equations (1) and (2) simultaneously.
Multiplying equation (1) by 4, we get:
-4m + 4g = 4 .........(3)
Adding equation (3) to equation (2), we get:
-4m + 4g + (-8m + 4g) = 4 + (-9)
-12m + 8g = -5 .........(4)
Dividing equation (4) by 4, we get:
-3m + 2g = -5/4 .........(5)
Now we have a system of equations (1) and (5):
-m + g = 1
-3m + 2g = -5/4
To solve this system, we can multiply equation (1) by 2 and subtract it from equation (5):
2(-m + g) = 2(1)
-2m + 2g = 2
-3m + 2g - (-2m + 2g) = -5/4 - 2
-3m + 2g + 2m - 2g = -5/4 - 8/4
-m = -23/4
Dividing both sides by -1, we get:
m = 23/4
Substituting the value of m into equation (1), we can find g:
-23/4 + g = 1
Adding 23/4 to both sides, we get:
g = 1 + 23/4
g = 4/4 + 23/4
g = 27/4
Therefore, the values of m and g are:
m = 23/4
g = 27/4
To find the values of m and g, we need to use the Remainder Theorem. According to the theorem, if a polynomial f(x) is divided by (x-a), then the remainder is equal to f(a).
Given that the remainder when f(x) is divided by (x+1) is 3, we can write:
f(-1) = 3
Substituting x = -1 into the given polynomial:
f(-1) = m(-1)^3 + g(-1)^2 - (-1)
= -m + g + 1
So, we have the equation:
-m + g + 1 = 3 .....(1)
Similarly, given that the remainder when f(x) is divided by (x+2) is -7, we can write:
f(-2) = -7
Substituting x = -2 into the given polynomial:
f(-2) = m(-2)^3 + g(-2)^2 - (-2)
= -8m + 4g + 2
So, we have the equation:
-8m + 4g + 2 = -7 .....(2)
Now, we can solve equations (1) and (2) simultaneously to find the values of m and g.
Taking equation (1), we have:
-m + g + 1 = 3
Rearranging the terms:
-m + g = 2 .....(3)
Taking equation (2), we have:
-8m + 4g + 2 = -7
Rearranging the terms:
-8m + 4g = -9 .....(4)
Now, we can solve equations (3) and (4) simultaneously. Here's how:
Multiply equation (3) by 4:
4(-m + g) = 4(2)
-4m + 4g = 8 .....(5)
Now, add equations (4) and (5):
-8m + 4g + (-4m + 4g) = -9 + 8
-12m + 8g = -1 .....(6)
Divide equation (6) by 4:
(-12m + 8g)/4 = -1/4
-3m + 2g = -1/4 .....(7)
Now, we have a system of equations (5) and (7) that we can solve simultaneously.
Multiply equation (5) by 3:
3(-4m + 4g) = 3(8)
-12m + 12g = 24 .....(8)
Now, add equations (7) and (8):
(-3m + 2g) + (-12m + 12g) = -1/4 + 24
-15m + 14g = 95/4
Simplifying:
-15m + 14g = 95/4 .....(9)
Now we have equation (9) that relates m and g.
To solve for m and g, we can multiply equation (3) by 3:
3(-m + g) = 3(2)
-3m + 3g = 6 .....(10)
Now, add equations (9) and (10):
(-15m + 14g) + (-3m + 3g) = 95/4 + 6
-18m + 17g = 119/4
Simplifying:
-18m + 17g = 119/4 .....(11)
Now, we have equation (11) that relates m and g.
To solve equations (9) and (11) simultaneously, we can use the method of elimination.
Multiply equation (9) by 2:
2(-15m + 14g) = 2(95/4)
-30m + 28g = 95/2 .....(12)
Multiply equation (11) by 3:
3(-18m + 17g) = 3(119/4)
-54m + 51g = 357/4 .....(13)
Now, subtract equations (12) and (13):
(-30m + 28g) - (-54m + 51g) = (95/2) - (357/4)
-30m + 28g + 54m - 51g = 190/4 - 357/4
24m - 23g = -167/4
Simplifying:
24m - 23g = -167/4 .....(14)
Now we have equation (14) that relates m and g.
We have a system of equations (10) and (14) that we can solve simultaneously.
Multiply equation (10) by 24:
24(-3m + 3g) = 24(6)
-72m + 72g = 144 .....(15)
Multiply equation (14) by 3:
3(24m - 23g) = 3(-167/4)
72m - 69g = -501/4 .....(16)
Now, add equations (15) and (16):
(-72m + 72g) + (72m - 69g) = 144 + (-501/4)
3g = 576/4 + (-501/4)
3g = 75/4
Divide both sides by 3:
g = (75/4)/3
g = 75/12
g = 25/4
Using equation (3), substitute the value of g:
-m + (25/4) = 2
Rearranging the terms:
-m = 2 - (25/4)
-m = 8/4 - 25/4
-m = -17/4
Multiply both sides by -1:
m = -(-17/4)
m = 17/4
Therefore, the values of m and g are:
m = 17/4
g = 25/4
To find the values of m and g, we can use polynomial division and the remainder theorem.
First, let's divide the function f(x) by (x + 1) and obtain the remainder 3:
(3x^3 + 2x^2 + gx - x) / (x + 1) = Q(x) + 3 / (x + 1)
This means that when we substitute x = -1 into f(x), the remainder is 3. Let's plug in x = -1 into the equation f(x) and solve for g:
f(-1) = (3(-1)^3 + 2(-1)^2 + g(-1) - (-1)) = -3 + 2 + g + 1 = 0 + g = g
Since the remainder is 3, we have:
3 = Q(-1) + 3 / (-1 + 1) = Q(-1)
Now, we divide f(x) by (x + 2) and obtain the remainder -7:
(3x^3 + 2x^2 + gx - x) / (x + 2) = P(x) + (-7) / (x + 2)
This means that when we substitute x = -2 into f(x), the remainder is -7. Let's plug in x = -2 into the equation f(x) and solve for g:
f(-2) = (3(-2)^3 + 2(-2)^2 + g(-2) - (-2)) = - 24 + 8 - 2g + 2 = -24 + 8 - 2g + 2
Simplifying further, we get:
-24 + 10 - 2g = -16 - 2g
Since the remainder is -7, we have:
-7 = P(-2) + (-7) / (-2 + 2) = P(-2)
Now, substituting the remainder values we obtained into the equations g = 0 + g and -16 - 2g = -7, we can solve for g:
g = 0 + g
-16 - 2g = -7
From the first equation, we see that g can be any value. However, from the second equation, we can solve for g:
-16 - 2g = -7
-2g = 9
g = -4.5
Thus, we have found the value of g. Since we only have one equation, we can't determine the value of m. Therefore, the value of m can be any real number.