What are the intercepts of F(x)=x^3-2x-6 and how do you find the answer?
HELP!
Do you mean the x-intercepts where y = 0? That would require solving for roots of the equation x^3 - 2x - 6 = 0
There is only one root, and it is at x = 2.1799810721581574...
Basically, you have to get it by trial and error, graphing or using a very long procedure that would fill two pages. I used a website for solving cubics. Google: cubic equation solver for examples.
The y-intercept, where x = 0, is at
F(x) = -6
To find the intercepts of the given polynomial function F(x), which is F(x) = x^3 - 2x - 6, you need to identify the points where the graph of the function intersects the x-axis or the y-axis.
1. To find the x-intercepts, set F(x) equal to zero and solve for x:
F(x) = x^3 - 2x - 6 = 0
This equation represents finding the values of x where the function crosses or touches the x-axis.
2. Unfortunately, the equation F(x) = x^3 - 2x - 6 is not easily solvable by factoring. So, we need to use other methods to find the real roots. One popular method is the Rational Root Theorem.
3. The Rational Root Theorem states that if a rational root exists, it will be in the form of a fraction p/q, where p is a factor of the constant term (-6) and q is a factor of the leading coefficient (1). In this case, the possible rational roots are ±1, ±2, ±3, ±6.
4. Use synthetic division or polynomial long division to divide the polynomial by each of the possible rational roots.
For example, let's try x = 1:
1 | 1 0 -2 -6
|______1___1__-1
1 1 -1 -7
The result of the division is 1x^2 + x - 7.
5. Repeat the process by trying each of the remaining possible rational roots until you find the real roots.
Continuing from the previous step, let's try x = -1:
-1 | 1 1 -1 -7
|______-1__0___1
1 0 -1 -6
The result of the division is 1x^2 - 6.
As we can see, x = 1 and x = -1 are two real roots of the function.
6. Set each real root equal to zero to find the x-intercepts:
x = 1 and x = -1
Therefore, the x-intercepts of F(x) = x^3 - 2x - 6 are x = 1 and x = -1.
7. To find the y-intercept, substitute x = 0 into the given function F(x):
F(0) = 0^3 - 2(0) - 6 = -6
Therefore, the y-intercept is when x = 0, y = -6.
In summary, the intercepts of F(x) = x^3 - 2x - 6 are:
- x-intercepts: x = 1 and x = -1
- y-intercept: y = -6