User:Aabicus/proof

The Original problem goes as such: Given: $$f(x) = x^4 - 3x^3 + 2x^2 - 7x - 11$$ Find all roots to the nearest .001

User:OdedSchramm plugged the equation into Mathematica, and revealed that there are two real answers and two complex answers. But I had to find them myself.

The first thing I did was graph this function on a graphing calculator, getting the zeros at (-.9112938,0) and (3.3290176,0). Therefore, these are my two "real" solutions. I still need to find the complex solutions.

The next thing I did was divide my original polynomial P by the polynomial (x + .9112938)(x − 3.3290176), which I multiplied out to make x² -2.4177238x -3.033766319. I then used synthetic division like such:


 * $$\begin{matrix}

2.4177238 & 3.033766319 & | & 1 & -3& 2 & -7 & -11 \end{matrix}$$

-


 * $$\begin{matrix}

2.4177238 & 3.033766319 & | & \underline{1} & -3& 2 & -7 & -11 \\ & & | & & 2.4177238& 3.033766319 & \\ & & | & & -.5822762& 5.033766319 & -7 & -11 \\ \end{matrix}$$

-


 * $$\begin{matrix}

2.4177238 & 3.033766319 & | & \underline{1} & -3& 2 & -7 & -11 \\ & & | & & 2.4177238& 3.033766319 & \\ & & | & & \underline{-.5822762} & 5.033766319 & -7 & -11 \\ & & | & & & -1.407783027 & -1.766489924 \\ & & | & & & 3.62593292& -8.766489924 \\ \end{matrix}$$

-


 * $$\begin{matrix}

2.4177238 & 3.033766319 & | & \underline{1} & -3& 2 & -7 & -11 \\ & & | & & 2.4177238& 3.033766319 & \\ & & | & & \underline{-.5822762} & 5.033766319 & -7 & -11 \\ & & | & & & -1.407783027 & -1.766489924 \\ & & | & & & \underline{3.62593292} & -8.766489924 \\ & & | & & & & 8.766626103& 11.00023317 \end{matrix}$$

For some reason the last bit of this final matrix won't work, so I'll write it out: -8.766489924 + 8.766626103 = .0000136179. -11 + 11.00023317 = .000023317 . It's good that the remainder is so small; it shows that my approximate answers were close.

The underlined numbers correspond to the coefficients of the quotient, the remaining numbers in the bottom row correspond to the coefficients of the remainder. The terms are written with increasing degree from right to left, with the last terms of the quotient and remainder each having degree 0.

So, the result of our division is
 * $$\frac{x^4 - 3x^3 + 2x^2 - 7 - 11}{x^2 - 2.4177238x - 3.033766319} = x^2 - .5822762x + 3.62593292 + \frac{.0000136179x + .000023317}{x^2 - 2.4177238x - 3.033766319}.$$

Since the roots of x² -.5822762x +3.62593292 are approximations of the roots of the original (we just drop the remainder), we can plug this polynomial into the quadratic formula getting:

$$x = \frac{.5822762 \pm \sqrt {-.5822762^2-4(3.62593292)}}{2},$$ - $$x = \frac{.5822762 \pm \sqrt {.3390455731-14.50373168}}{2},$$ - $$x = \frac{.5822762 \pm \sqrt {-14.16468611}}{2},$$ - $$x = \frac{.5822762 \pm \sqrt {14.16468611}i}{2},$$ - $$x = \frac{.5822762 \pm {3.763600153i}}{2},$$ - $$x = {.2911381 \pm {1.881800076i}},$$

Since the plus-minus represents two answers, one positive and one negative, I have my two complex answers.

The four roots are .912, .320 , .291+1.882i , and .291-1.882i.

Note:Extra Special thanks to User:OdedSchramm and User:Tango, who were invaluable in helping me understand all this.