f(x) is a cubic polynomial such that f(n)=1/n^2+1 for n=1,2,3,4. If f(0)=a/b,where a and b are coprime positive integers, what is the value of a+b
To find the value of a and b, we need to determine the value of f(0) using the given information.
Since f(x) is a cubic polynomial, its general form can be written as f(x) = ax^3 + bx^2 + cx + d, where a, b, c, and d are constants.
We are given that f(n) = 1/n^2 + 1 for n = 1, 2, 3, 4. Let's substitute these values into the polynomial:
f(1) = a(1^3) + b(1^2) + c(1) + d = 1/1^2 + 1 = 2
f(2) = a(2^3) + b(2^2) + c(2) + d = 1/2^2 + 1 = 5/4
f(3) = a(3^3) + b(3^2) + c(3) + d = 1/3^2 + 1 = 10/9
f(4) = a(4^3) + b(4^2) + c(4) + d = 1/4^2 + 1 = 17/16
We have four equations:
a + b + c + d = 2
8a + 4b + 2c + d = 5/4
27a + 9b + 3c + d = 10/9
64a + 16b + 4c + d = 17/16
We can solve these simultaneous equations to find the values of a, b, c, and d. Let's use the method of elimination.
Subtracting the first equation from the second, third, and fourth equations, respectively, we get:
7a + 3b + c = -3/4
26a + 8b + 2c = 6/9
63a + 15b + 3c = 15/16
We can multiply the first equation by 2, the second equation by 3, and the third equation by 16 to simplify the system:
14a + 6b + 2c = -6/4
78a + 24b + 6c = 18/9
1008a + 240b + 48c = 15
Subtracting the first equation from the second, we get:
64a + 18b + 4c = 12/9
Subtracting the first equation from the third, we get:
994a + 234b + 46c = 21
Now, let's subtract two times the expression we just obtained from the expression we got after subtracting the first equation from the third:
(994a + 234b + 46c) - 2(64a + 18b + 4c) = 21 - 2(12/9)
Simplifying, we have:
994a + 234b + 46c - 128a - 36b - 8c = 21 - 24/9
Combining like terms:
866a + 198b + 38c = 239/9
Now we have two equations:
64a + 18b + 4c = 12/9
866a + 198b + 38c = 239/9
Multiplying the first equation by 38 and the second equation by 4 to eliminate the term with c, we get:
2432a + 684b + 152c = 456/9
3464a + 792b + 152c = 956/9
Subtracting the first equation from the second, we have:
1032a + 108b = 500/9
To solve for a and b, we need one more equation. Let's use f(0) = a/b.
f(0) = d = a(0^3) + b(0^2) + c(0) + d = d
We can use d as the third equation:
d = a/b
Now we have three equations:
1032a + 108b = 500/9
2432a + 684b = 456/9
d = a/b
To find the values of a and b, we need to solve this system of equations further. Let's proceed with the next steps.
f(x) = ax^3 + bx^2 + cx + d
f(1) = 1/2
f(2) = 1/5
f(3) = 1/10
f(4) = 1/17
a+b+c+d = 1/2
8a+4b+2c+d = 1/5
27a+9b+3c+d = 1/10
64a+16b+4c+d = 1/17
f(x) = 1/170 (-4x^3 + 41x^2 - 146x + 194)