f(x) is a cubic polynomial that satisfies f(1)=f(2)=f(3)=4 and f(0)=0. What is the value of f(7)?
f(x) = ax^3 + bx^2 + cx
a+b+c = 4
8a+4b+2c = 4
27a+9b+3c = 4
f(x) = 2x/3 (x^2 - 6x + 11)
f(7) = 84
Thanks Steve!
To find the value of f(7), we need to determine the cubic polynomial that satisfies the given conditions.
A cubic polynomial has the general form f(x) = ax^3 + bx^2 + cx + d, where a, b, c, and d are constants.
First, let's use the fact that f(0) = 0 to find the value of d. Substituting x = 0 into the equation, we have:
f(0) = a(0^3) + b(0^2) + c(0) + d
0 = 0 + 0 + 0 + d
d = 0
Next, let's use the fact that f(1) = f(2) = f(3) = 4 to set up a system of equations. Substituting these values into the equation, we get:
f(1) = a(1^3) + b(1^2) + c(1) + 0 = 4
f(2) = a(2^3) + b(2^2) + c(2) + 0 = 4
f(3) = a(3^3) + b(3^2) + c(3) + 0 = 4
Simplifying each equation, we have:
a + b + c + 0 = 4 (equation 1)
8a + 4b + 2c + 0 = 4 (equation 2)
27a + 9b + 3c + 0 = 4 (equation 3)
From equation 1, we can express a in terms of b and c:
a = 4 - b - c
Substituting this expression into equations 2 and 3, we get:
8(4 - b - c) + 4b + 2c + 0 = 4
27(4 - b - c) + 9b + 3c + 0 = 4
Simplifying these equations will give us the specific values of b and c.
Now, let's solve this system of equations to find the values of b and c:
32 - 8b - 8c + 4b + 2c = 4
108 - 27b - 27c + 9b + 3c = 4
Combine like terms:
-4b - 6c + 32 = 4
-18b - 24c + 108 = 4
Rearrange the equations:
-4b - 6c = -28 (equation 4)
-18b - 24c = -104 (equation 5)
Next, solve equations 4 and 5 simultaneously to obtain the values of b and c.
Multiply equation 4 by 9 and equation 5 by 2:
-36b - 54c = -252 (equation 6)
-36b - 48c = -208 (equation 7)
Subtract equation 7 from equation 6:
6c = -44
c = -44/6
c = -22/3
Substituting this value of c back into equation 4:
-4b + 22/3 = -28
-4b = -28 - 22/3
-4b = -84/3 - 22/3
-4b = -106/3
b = (106/3)/(-4)
b = -106/12
b = -53/6
Now that we have the values of b and c, we can find the value of a by plugging them into equation 1:
a + (-53/6) + (-22/3) = 4
a = 4 + 53/6 + 22/3
a = (24/6) + (53/6) + (44/6)
a = 121/6
So, the cubic polynomial that satisfies the given conditions is f(x) = (121/6)x^3 - (53/6)x^2 - (22/3)x.
Finally, to find f(7), we substitute x = 7 into the equation:
f(7) = (121/6)(7^3) - (53/6)(7^2) - (22/3)(7)
Evaluating this expression will give us the value of f(7).