Suppose f is a polynomial such that
f(0)=47, f(1)=32,f(2)=-13,
and f(3)=16. what is the sum of the coefficients of f?
First, note that the degree of f must be 3. If higher, there are many possible polynomials which will fit the four points.
If f(x) = ax^3+bx^2+cx+d
d = 47
a+b+c+d = 32
8a+4b+2c+d = -13
27a+9b+3c+d = 16
a+b+c+d = 32
all I know is that it's NOT -15... believe me I tried lol
To find the sum of the coefficients of the polynomial f, we need to determine the degree of the polynomial. Since we are given four distinct values for f(x), the degree of the polynomial will be at most 3.
Let's assume the polynomial f(x) has the form:
f(x) = ax^3 + bx^2 + cx + d
Now, we can substitute the given values into the equation to form a system of equations:
f(0) = a(0)^3 + b(0)^2 + c(0) + d = d = 47
f(1) = a(1)^3 + b(1)^2 + c(1) + d = a + b + c + d = 32
f(2) = a(2)^3 + b(2)^2 + c(2) + d = 8a + 4b + 2c + d = -13
f(3) = a(3)^3 + b(3)^2 + c(3) + d = 27a + 9b + 3c + d = 16
Now we have a system of four equations with four unknowns (a, b, c, d). We can solve this system of equations to find the values of a, b, c, and d.
Subtracting equations (1) and (2), we get:
(8a + 4b + 2c + d) - (a + b + c + d) = -13 - 32
7a + 3b + c = -45
Subtracting equations (2) and (3) gives:
(27a + 9b + 3c + d) - (8a + 4b + 2c + d) = 16 - (-13)
19a + 5b + c = 29
Now, we have a system of two equations with two unknowns (a, b):
7a + 3b + c = -45
19a + 5b + c = 29
We can solve this system of equations. Subtracting the first equation from the second equation, we get:
19a + 5b + c - (7a + 3b + c) = 29 - (-45)
12a + 2b = 74
Simplifying further, we have:
12a + 2b = 74
6a + b = 37
Now, we can solve this system of equations using substitution or elimination to find the values of a and b.
By solving this system of equations, we find that a = 3 and b = 19.
Now, let's substitute the values of a and b back into one of the earlier equations to find c:
7a + 3b + c = -45
7(3) + 3(19) + c = -45
21 + 57 + c = -45
78 + c = -45
c = -45 - 78
c = -123
Finally, to find the sum of the coefficients of f, we add the coefficients:
a + b + c + d = 3 + 19 + (-123) + 47 = -54
Therefore, the sum of the coefficients of f is -54.