First I will state the roots of the function as follow
And then prove this using the Remainder Theorem:
is a root of
Substituting x with the values of the roots:
Verification on the calculator :
The next step will be to find the equation of the tangent line of two of the three roots. The first thing to do when finding the equation of a tangent line is to find the slope (gradient). Hence to find the derivative of the function
and
Second is to find the average of two of the roots using the formula, (a and b being the roots) and then substitute x with the value of the average in the function as following:
Average =
This implies that the slope at (-2.25, 4.22) will be:
I will use the formula of the equation of a line and rewriting it to find which is the equation of the tangent. * is the average of two of the roots respectively ; is and is .
In this case, and
The same steps I used for the average of the other roots:
Average =
Slope at (-0.75, -7.6) will be:
Equation of the tangent where, and
Average =
Slope at (0, -13.5) will be:
Equation of the tangent where, and
The next thing to do is to find out where the tangents lines will intersect with the curve again. For that to occur the equation of the tangent lines should be equal with 0 ().
The intersection will be at (1.5, 0)0
The intersection will be at (-1.5, 0)
The intersection will be at (-3.0, 0)
As I observe in the graph and in my calculations the tangent line at the average of two of the roots will intersect the graph at the other root. An observation which I can base it on my conjecture: The equation of a tangent at the average of any two roots will intersect the y-axis at the third root.
The next step will be to test my conjecture in another similar cubic function using the same steps as previous.
Let us consider the function:
The graph of the function:
*Window values:
* The average:
* The equation of tangent:
* The slope () of the function
*
The roots of the function are:
Proof using the Remainder Theorem:
Verification on the calculator :
Average
Slope at (-0.25, 2.53) will be:
Equation of the tangent where, and
Intersection occurs when the intersection will be at (2, 0)
Average
Slope at (0.5, 0) will be:
Equation of the tangent where, and
Intersection occurs when the intersection will be at (0.5, 0)
Average
Slope at (1.25, -2.53) will be:
Equation of the tangent where, and
Intersection occurs when the intersection will be at (-1, 0)
Observing from my calculations the equation of the tangent line to the average of two of the three roots will intersect the graph at the third root.
With this I can conclude that the conjecture stands for other functions as well.
Now to prove my conjecture I will use the same steps and formulas but this time I will not use values but variables to have a general formula.
where, b and c are the roots and k is a constant that changes the position of the curve. To simplify my calculation I will assume that
This could be written also as:
*this might be useful when finding the derivative of the function in an easier way
The average of two of the roots will be .
The slope will be :
To find the equation of the tangent I will use the previous formulas:
*
*
*
*
As when the tangent cuts the x-axis
As x = c, which is the third i.e. the root not included in the average, we can conclude that the conjecture is true for all cubic equations. Hence, the equation of tangent at the average of any two roots will intersect the y–axis at the third root.
Forward I will investigate the above properties with cubic functions which have one root or two roots.
I will first start with the cubic function which has only one root:
Let us consider the function:
*Window values:
The root of the equation is:
Proof using the remainder theorem:
Equation of the tangent line
-
Roots -1.1456 and -1.1456
Average
Slope at (-1.1456, 0) will be:
Equation of the tangent where and at
This shows that the conjecture is applicable to a cubic equation with one root, although not required.
Now I will investigate a cubic function with two roots.
Let us consider the function:
*Window values:
The roots of the function are:
Proof using the remainder theorem:
Equation of the tangent line (average of two of roots):
Slope at (-1.5, 0.125) will be:
Equation of the tangent where and at y = 0,
This again shows that the conjecture is applicable to a cubic equation with two roots.
With this I will conclude that the conjecture stated is valid.