Solving Polynomials

My mom's friend has requested that I show how to solve polynomials so this page provides an instruction set on how to solve for polynomials using several different techniques. Some of the techniques you may use (although not all are included here as instruction sets) are:

  • Using Solver Function in Excel.
  • Using Solver Function in TI-83.
  • Graphing in Excel.
  • Graphing in T1-83 and using Find Root Option.
  • Use Another Computer Program such as Mathematica or Matlab.
  • Use Newton's Method.
  • Use Algebraic Tricks if it is a Simple Polynomial.

 

I will now discuss three ways that you can solve for the roots of a polynomial equation.


Using Solver in Excel

Excel is a frequently computer program that has many uses. One of the most powerful tools that Excel has is the Solver tool. By following this instruction set, you will learn how to use the Solver add-in to find the solution to a polynomial equation. To complete this instruction set, you will need:

  •    Microsoft Excel 2007.

  •   Approximately 3-4 minutes of free time.

  •   Desired Equation to be solved.

 

Follow the instruction set below to solve equations using Excel. You can also follow along with the video screen cast shown below. This procedure can be used with older versions of Microsoft Excel but the step-by-step instructions use Microsoft Excel 2007.

 

1) Open Microsoft Excel on your computer.


2) Obtain your desired equation and use algebraic manipulation to have the equation equal zero on one side of the equation.


solve_2.png


3) Create a cell for each coefficient in the spreadsheet.


solver3.png


4) Create a cell for your given variable and also make an initial guess.


solver_4.png


5) Create a cell for your equation.

Note: Be sure to use the equals sign before typing in the equation into the formula bar. If you already have the Solver add-in, go to Step 10.


solver5.png


6) Go to the window button in the top left corner of the screen.


solver6.png


7) Click "Excel Options" at the bottom of the drop down menu.


repair.png



8) Select "Add-Ins" from the menu to the left hand side of the pop up screen.


solver8.png


9) Select "Solver Add-In" underneath "Active Application Add-Ins".


solver9.png



10) Go to the "Data" tab at the top of the screen.


solver10.png



11) Open up the Solver application to the right side of the screen.


solver11.png



12) In the pop up window, set the target cell as your equation cell.


solver12.png



13) In the same window, set the target cell to a value of 0.


solver13.png



14) In the same window, allow your variable cell to be the cell that changes.


solver14.png



15) Click "Solve".

Note:  Another pop up window should appear telling you that a solution was found.  


solver15.png



16) Click "OK" in the popup window, and your answer should appear in the variable cell.


solver16.png




Congratulations! You have now successfully solved the polynomial equation. The answer is exact to approximately the thousandths place.


Video can be seen here.

 

 

 

Graphing in Excel

Microsoft Excel can also be used in a more straightforward sense to find the zeros to a polynomial equation. This can be accomplished by plotting the function at many different values and analyzing how the height of the function changes as the independent variable changes. The method can be less efficient and harder to determine where the zeros of the equation lie, but it is an easier approach to finding the roots of the equation. By following this instruction set, you will be able to find the roots of an equation by plotting the function at different discrete values. In order to complete this instruction set, you will need:

  • Microsoft Excel 2007.

  • Approximately 3-4 minutes of free time.

  • Desired Equation to be solved.

 

 

Follow the instruction set below to solve equations using Excel. You can also follow along with the video screen cast shown below. This procedure can be used with older versions of Microsoft Excel but the step-by-step instructions use Microsoft Excel 2007.

 

 

1) Open Microsoft Excel in your computer.


2) Obtain your desired equation and use algebraic manipulation to have the equation equal zero on one side of the equation.


solve_2.png



3) Create a cell for each coefficient in the spreadsheet.


solver3.png



4) Create a column for the variable in the spreadsheet.


graph4.png



5) Create a column for the equation in the spreadsheet.


graph5.png



6) Place your initial value to evaluate the function into the first row of your variable column.


graph6.png



7) Type in the equation into the formula bar for the variable in the same row.

Note: Be sure to use an equals sign before entering the actual equation into the formula bar. Type the equation in terms of the variable cell and the coefficient cells. Make certain that you put '$' before the letter and number of the cell for each coefficient to ensure that the coefficients do not change as you change your variable values. If you do not do this then you will have an error later in this instruction set.


graph7.png



8) Create your time step by picking a small number and adding it to the initial value. A typical time step value should be around 0.01 or 0.001.


graph8.png



9) Use the drag down feature in Excel to create a value of the equation for the initial value plus the value of the time step.

Note: This is accomplished by clicking the initial equation cell, clicking the small black box at the bottom right hand corner of the highlighted cell, and dragging down.


graph9.png




10) Use the drag down feature again by highlighting all four cells that you have created and dragging down.

Note: Continue to drag down until you have reached either a very high value or the value at the end of the range that you are examining the function at.


graph10.png




11) You can now examine where the function values cross from negative to positive values in the equation column. This indicates that you have found a root for the equation.

Note: You may also graph the values by using a scatter plot. This process can be used to see what the function is doing visually. This process can be seen by watching the video tutorial. You can select all the data points and graph, and you may also edit what you are graphing by right clicking the graph and choosing "Select Data". However, graphing is not necessary to find the root to the equation- it is simply another feature that can be used when solving for the roots of the equation in this manner.


graph11.png


 

Congratulations! You have now successfully solved for the roots of the polynomial equation.


Video can be seen here.

 

 

 

Using Newton's Method

If you do not feel comfortable solving in Microsoft Excel, there is another alternative for finding the roots of the equation. The way that this can be accomplished is by using Newton's Method. This process relies on making an initial guess and iterating over several values. While this process is possible to complete without the use of a calculator, it is not recommended. In order to use Newton's Method, you need to be able to calculate the derivative of your equation. This is not a difficult task when working with polynomials (Take the exponent of the variable and multiply this number by the coefficient for the variable to the designated power and then subtract one from the exponent of the variable). By following this instruction set, you will be able to solve for the zeros of a polynomial equation by using Newton's Method. In order to complete this instruction set, you will need:

  • A calculator.

  • Knowledge of finding the derivative of an equation.

  • 3-10 minutes of free time.

 

Follow the instruction set below to solve for the roots of the equation using Newton's Method.


1) Create a table with column names x0, f(x0), f'(x0), and x1.


newton1.png



2) Calculate the derivative of the equation, f'(x).


newton2.png



3) Create an initial guess and put this number in the first row under the x0 column.


newton3.png



4) Calculate f(x0) and f'(x0) at the value for the initial guess by plugging the initial guess into the function and into the derivative.


newton4.png



5) Calculate x1 by using the formula used during Newton's Method: x1= x0- (f(x0)/f'(x0)).

newton5.png

6) Place each of these values into the corresponding value in the table.


7) Your x1 value becomes the x0 value in the next row if the value of the function at the point is not sufficiently low.


8) Repeat this process until your function value is very low (approximately 0).

newton7.png



9) When your function value is approximately 0, find the x0 for that row and this is one of the roots for the equation.

newton_9.png

 

Congratulations! You have now successfully solved for the roots to the equation using Newton's Method.


Video can be seen here.






 

 

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