f(12)+f(−8)=25,344+11,264= 36,608.
We can solve this problem by using the given information to form a system of equations and then solving for the coefficients of the polynomial.
From the given information, we know that:
f(1)=10
f(2)=20
f(3)=30
Substituting these values into the equation f(x)=x^4 +ax^3 +bx^2 +cx+d, we get:
1+a+b+c+d=10
16+8a+4b+2c+d=20
81+27a+9b+3c+d=30
We can rewrite this system of equations as a matrix equation:
| 1 | 1 | 1 | 1 | 1 | 10 |
| 16 | 8 | 4 | 2 | 1 | 20 |
| 81 | 27 | 9 | 3 | 1 | 30 |
Using Gaussian elimination, we can solve for the coefficients of the polynomial. The solution is:
a=6
b=−2
c=13
d=−10
Therefore, the polynomial is f(x)=x^4 +6x^3 −2x^2 +13x−10.
Now we can find the value of f(12)+f(−8):
f(12)=12^4 +6(12)^3 −2(12)^2 +13(12)−10=25,344
f(−8)=(−8)^4 +6(−8)^3 −2(−8)^2 +13(−8)−10=11,264
Therefore, f(12)+f(−8)=25,344+11,264= 36,608.
Question
If `f(x)=x^4+ ax^3 + bx^2 + cx + d and f(1)=10, f(2)=20, f (3)=30` then find the value of `f (12)+f(-8)`.