Maple Tutor

Part 11: Algebraic Operations

In this part we examine algebraic operations that are useful in dealing with polynomials and quotients of polynomials, i.e., rational functions.

1. If necessary, unassign x:
   x:='x'
   Then enter the polynomial
   P:=3*x^5-18*x^4-7*x^3+42*x^2-40*x+240

   Use your mouse to highlight this expression, click the right mouse button, and select "Factor" from the popup menus.

   Maple will not break quadratic factors into linear ones if the corresponding roots are not rational, e.g., square roots or complex numbers with nonzero imaginary parts.

2. We may use the solve menu option to obtain a complete factorization. Enter the following
   P=0
   and press RETURN.  You should see
   

   3x⁵-18x⁴-7x³+42x²-40x+240=0
   
   

   as the output.
   

   Use your mouse to highlight this expression, click the right mouse button, and select "Solve"  > "Obtain Solutions for" > "x" from the popup menus. (Note that this is not quite the same menu option we chose in Part 2.)
   

   Notice in the complex roots that Maple uses a capital I for the imaginary unit -- rather than just i.

3. Next we convert our answer into a Maple list and give it a name. Use your mouse to highlight your solutions, click the right mouse button, and select "Conversions" > "To List" from the popup menus.  Notice that Maple lists have square brackets around them.
   
   Now use your mouse to highlight the list (including the square brackets) , click the right mouse button, and select "Assign to a Name" from the popup menus.  In the box that appears, enter "rootList".
   

   We may check our work by multiplying the linear factors together with a scalar factor of 3. We will use the palettes again.  Enter
   3*
   and then click on the palette button which says "Expression", and select the capital pi (for "Product") symbol from the grid of symbols which appears.  (If you are not familiar with capital-pi notation for products, it works just like the capital-sigma notation for sums.)  Maple will insert a product template into the document with places to enter the index of the product, the upper and lower bounds, and the formula for the factors.
   
   

   1. The cursor is positioned for you to enter the index for the product. Enter k.
      
      
   2. Press the Tab key. Notice that the cursor moves to the lower bound. Enter 1, then Tab, then 5, then Tab again, and finally (x - rootList[k]).
      
      
   3. Now, if you press Enter, Maple will construct a product.
      
      Use your mouse to highlight the product, click the right mouse button, and select "Expand" from the popup menus.  Did the polynomial return to its original form?
      

4. Enter a new polynomial
   P2:=x^7+6*x^6-3*x^5-x^4+2*x^3-x^2+5*x-1;

   What happens when you try the factor and solve menu options on this polynomial?

   Maple is not always able to determine the exact roots of a polynomial of degree greater than four.  We may also obtain approximate roots using the solve menu option. Enter
   P2=0
   and press RETURN.  Use your mouse to highlight the output, click the right mouse button, and select "Solve"  > "Numerically Solve" from the popup menus.

   Notice that this command only returned three real roots. The other roots have nonzero imaginary parts. To obtain them, we must deal with each root individually.  Enter
   P2=0
   and press RETURN again.  Use your mouse to highlight the output, click the right mouse button, and select "Solve"  >  "Solve"  > "Obtain Solutions for" > "x" from the popup menus. You should get a sequence of expressions, each starting with RootOf.  Copy and paste one of these expressions onto a new line.  Then highlight it, click the right mouse button, and select "Approximate"  > "10" from the popup menus.  Do this for each of the other expressions until you have seven different approximate roots.
   
   

5. Now we turn to operations on quotients of polynomials. Enter the following rational expression
   Q := 2/(1+3*x) + x/(1 + x^2) + 1/(2 - x)

   Put Q in "normal" form by highlighting it, clicking the right mouse button, and selecting "Normal" from the popup menus.
   

   "Normal" for Maple means writing the rational expression as a quotient with all common factors canceled.

   Suppose you want to expand out the denominator. First pick off the numerator and denominator of the polynomial by highlighting it, clicking the right mouse button, and selecting "Numerator" from the popup menus. Then do the same thing, but click "Denominator".
   
   

   Now expand out the denominator by highlighting it, clicking the right mouse button, and selecting "Expand" from the popup menus.
   

   Now reassemble the expanded expression on a new line using copy and paste.
   

6. Now we go in the other direction. If we expand the new expression in partial fractions, we should wind up back where we started. Highlight the new expression, click the right mouse button, and select  "Conversions" > "Partial Fractions" > "x".
   

   Is this what you started with?  Is it equivalent to what you started with?