Solving Polynomials Equations
of Higher Degree

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When the powers in polynomial equations increase, it becomes more difficult to find their solutions (roots). 

Consider an equation such as  
                

Finding the roots of an equation such as this can prove to be quite a task.  In this course, we will just be touching the surface on techniques for solving higher degree polynomial equations.

This course will examine how to solve polynomial equations of higher degree using factoring and/or the quadratic formula, only.


FYI:  The equation at the left factors into

Let's be sure that we understand the vocabulary associated with this type of task.

              The following statements are different ways of
                                   asking the same thing!!

    Solve the polynomial equation P(x) = 0.
    Find the roots of the polynomial equation P(x) = 0.
    Find the zeroes of the polynomial function P(x) (P(x) = 0).
    Factor the polynomial function P(x) = 0 and express the roots.

How many roots should we expect to find?  A polynomial of degree n will have n roots, some of which may be multiple roots (they repeat).  For example, is a polynomial of degree 3 (highest power) and as such will have 3 roots.  This equation is really giving solutions of x = 1 and x = 4 (repeated).


Examples:

1.  Solve the following polynomial equation: 
                       Solution Method:  Recognize a pattern within the problem.
 

  We are looking for 4 roots.

  Set the equation equal to 0.
  Notice that this problem is really the variable x2 being squared and being used to a power of one.  Get in the habit of looking for this pattern.

  Letting x2 = a may help you to see the rest of the solution more easily.  Make the substitutions.

  Now, we have a nice quadratic equation that we know how to solve.  This one factors nicely.

  Be careful NOT to STOP when you solve for a.  Remember that a really represents x2.

  Replace a with x2 and solve for the answers to the original equation.


 

2.  Find the roots of the polynomial equation .
                 Solution method:  Find common factor first then recognize a pattern.
 

  We are looking for 5 roots.

  There is a common factor of t.  Factor it out.

  This problem now contains the same pattern we saw in example 1.  It contains the variable t2 being squared and being used to a power of one.  Substitution of another letter is not being used is this example, but could be used if you wish.
  Factor the quadratic.

  Set each factor equal to zero and solve.  Be sure to list both the plus and minus versions when solving the t2 equations.

 

3.  Find the zeroes of the polynomial function (P(x) = 0) when
 
                             
                          Solution method:  Use the quadratic formula.

 

  We are looking for 3 roots.  Think about the highest power of x if the problem were multiplied out.

  Set each factor equal to zero.  The factor on the left needs to be factored further.  Unfortunately, this cannot be done easily. 

  Use the quadratic formula to find the roots from the first factor.

  Solve for x.