The polynomial f(x) has degree 3. If f(-1)=15, f(0)=0, f(1)=-5, f(2)=12, then what are the x-intercepts of the graph of f?
clearly, f(0) = 0
so,
f(x) = ax^3+bx^2+cx
= x(ax^2+bx+c)
-1(a-b+c) = 15
1(a+b+c) = -5
2(4a+2b+c) = 12
solve those and you wind up with
f(x) = 2x^3+5x^2-12x
See
http://www.wolframalpha.com/input/?i=table+2x^3%2B5x^2-12x+for+x+%3D+-1..2
Now just solve for the roots of f(x)
To find the x-intercepts of the graph of f(x), we need to find the values of x for which f(x) equals zero.
Since the degree of the polynomial f(x) is 3, it can be represented as:
f(x) = ax^3 + bx^2 + cx + d
Substituting the given values of f(x) at x = -1, 0, 1, and 2 into the equation, we can set up the following system of equations:
f(-1) = a(-1)^3 + b(-1)^2 + c(-1) + d = 15 -- (Eq. 1)
f(0) = a(0)^3 + b(0)^2 + c(0) + d = 0 -- (Eq. 2)
f(1) = a(1)^3 + b(1)^2 + c(1) + d = -5 -- (Eq. 3)
f(2) = a(2)^3 + b(2)^2 + c(2) + d = 12 -- (Eq. 4)
Simplifying these equations gives us:
-a + b - c + d = 15 -- (Eq. 1 simplified)
d = 0 -- (Eq. 2 simplified)
a + b + c + d = -5 -- (Eq. 3 simplified)
8a + 4b + 2c + d = 12 -- (Eq. 4 simplified)
Substituting d = 0 into (Eq. 2 simplified) gives us:
a + b + c = -5 -- (Eq. 5)
Substituting this back into (Eq. 1 simplified), we have:
-a + b - c = 15 -- (Eq. 6)
Adding (Eq. 5) and (Eq. 6) gives us:
2b = 10
b = 5
Substituting b = 5 into (Eq. 5), we can solve for a + c:
5 + a + c = -5
a + c = -10 -- (Eq. 7)
Substituting b = 5 into (Eq. 1 simplified) gives us:
-a + 5 - c = 15
-a - c = 10 -- (Eq. 8)
Adding (Eq. 7) and (Eq. 8) gives us:
2a = 0
a = 0
Substituting a = 0 into (Eq. 7), we have:
0 + c = -10
c = -10
Therefore, the polynomial f(x) can be written as:
f(x) = 0x^3 + 5x^2 - 10x + 0
f(x) = 5x^2 - 10x
Now, to find the x-intercepts, we set f(x) equal to zero and solve for x:
5x^2 - 10x = 0
Factoring out 5x:
5x(x - 2) = 0
Setting each factor equal to zero:
5x = 0 or x - 2 = 0
Solving for x gives us:
x = 0 or x = 2
Therefore, the x-intercepts of the graph of f are x = 0 and x = 2.
To find the x-intercepts of the graph of the polynomial function f(x), we need to find the values of x for which f(x) equals zero. Given that f(0) = 0, we can conclude that x = 0 is one of the x-intercepts.
Since the polynomial f(x) has a degree of 3, it can be written in the form:
f(x) = ax^3 + bx^2 + cx + d
To determine the values of a, b, c, and d, we'll use the given values of f(-1), f(1), and f(2).
1. Using f(-1) = 15, we substitute x = -1 into the equation:
15 = a(-1)^3 + b(-1)^2 + c(-1) + d
Simplifying, we get:
-15 = -a + b - c + d
2. Using f(1) = -5, we substitute x = 1 into the equation:
-5 = a(1)^3 + b(1)^2 + c(1) + d
Which simplifies to:
-5 = a + b + c + d
3. Using f(2) = 12, we substitute x = 2 into the equation:
12 = a(2)^3 + b(2)^2 + c(2) + d
Simplifying, we get:
12 = 8a + 4b + 2c + d
Now, we have a system of linear equations consisting of the equations obtained from steps 1, 2, and 3. Solving this system of equations will give us the values of a, b, c, and d.
Subtracting the equation obtained in step 2 from the equation obtained in step 1, we get:
-10 = -9a - 9b - 9c - 9d
Adding the equation obtained in step 2 to the equation obtained in step 3, we get:
7 = 9a + 5b + 3c + d
Combining these two equations, we have:
-10 + 7 = (-9a - 9b - 9c - 9d) + (9a + 5b + 3c + d)
-3 = -4b - 6c - 8d
Now, we can substitute the values of b and d in terms of c to obtain a single-variable equation:
-3 = -4b - 6(-2c - 2)
-3 = -4b + 12c + 12
Rearranging this equation, we get:
4b = 12c + 15
Dividing by 4, we have:
b = 3c + 3.75
Now, we can substitute this expression for b into one of the previous equations (e.g., the equation obtained in step 2: -5 = a + b + c + d) to obtain a single-variable equation:
-5 = a + (3c + 3.75) + c + d
-5 = a + 4c + 3.75 + d
Rearranging this equation, we get:
a = -4c - 8.75 - d
We now have expressions for a, b, and d in terms of c. Substituting these expressions into the equation -10 = -4b - 6c - 8d, we get:
-10 = -4(3c + 3.75) - 6c - 8d
-10 = -12c - 15 - 6c - 8d
-10 = -18c - 8d - 15
Rearranging this equation, we get:
18c + 8d = 15 - 10
18c + 8d = 5
Since there are infinitely many values of c and d that satisfy this equation, we can choose any convenient values and solve for the corresponding values of a and b. For simplicity, let's choose c = 0 and d = 0.
By substituting c = 0 and d = 0 into the expressions for a and b, we find:
a = -8.75
b = 3.75
Finally, we can write the polynomial function f(x) as:
f(x) = -8.75x^3 + 3.75x^2
To find the x-intercepts of the graph of f, we set f(x) to zero:
0 = -8.75x^3 + 3.75x^2
Next, we factor out common terms:
0 = x^2(-8.75x + 3.75)
This equation is satisfied if either x^2 = 0 or -8.75x + 3.75 = 0.
From x^2 = 0, we find that x = 0.
To find the other x-intercept, we solve -8.75x + 3.75 = 0 for x:
8.75x = 3.75
x = 3.75 / 8.75
x ≈ 0.43
Therefore, the x-intercepts of the graph of f are x = 0 and x ≈ 0.43.