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Real-Zeros-of-Polynomial-Functions

Page history last edited by Kristen Fouss 15 years, 3 months ago

2.3 Real Zeros of Polynomial Functions

 

 

Polynomial Function: F(x)= eq=ax^n + ax^n-1+...+g

 

To find real zeros of polynomial functions you can:

 

1. Divide by known factors (synthetic division/quadratic formula)

     - If x-a is a factor, factor it out of the equation by using synthetic division

     - If the remainder is zero, then the number is a factor

 

Example #1:

     Is x-eq=\frac{3}{2} a factor of eq=2x^3 -3x^2 -2x+3

 

     Yes, x-eq=\frac{3}{2}  is a factor of eq=2x^3 -3x^2 -2x+3  

  

- after factoring out x-eq=\frac{3}{2}, keep factoring until simplified

- solve the problem to find the zeros

 

Link to see more examples:

     http://id.mind.net/~zona/mmts/functionInstitute/polynomialFunctions/roots/syntheticDivision.html 

 

 

 

2. Rational Root Therom 

Rational Root Therom: All possible rational roots are of the for eq=\frac{p}{q}.

 

  

      p-- factors of last tern of the polynomial function

      q-- factors of the first term of the polynomial function

 

Example #2: 

     Find all possible rational roots in the equation F(x)= 

     1eq=x^4 + 2x^3 - 7x^2 - 8x + 12

 
 

      p-- eq=\frac{+}{}  1, 2, 3, 4, 6, 12 

 

      q-- eq=\frac{+}{}  1

 

 

eq=\frac{+}{} \frac{1,2,3,4,6,12}{1}

 

= –12, –6, –4, –3, –2, –1, 1, 2, 3, 4, 6, 12

 

 

For further explination visit http://www.purplemath.com/modules/rtnlroot2.htm

 

 

3. Graph

 

- Plug in the equation to the calculator

 

 

Example #3: Graph f(x)= eq=x^3 - 2x^2 - 5x + 6

      

 

 

x = -2, 1, 3

 

-zeros cross the x-axis

-to find zeros using calculator second trace (calc) zero

 

Example #4: Graph f(x)= eq=x^2 -x -2

 

Graph for problem 7; a right-side-up bowl 

 

x= -1, 2

 

Now it's your turn!

 

Is (x-1) a factor of the following?

 

1. f(x) = x4 - 2x3 + x - 2  

2. f(x) = x3 + 2x2 - 3x + 1

Use the Rational Root Therom to find out the rational roots.

1. f(x) = x4 - 2x3 + x - 2  

 

2. f(x) = x3 + 2x2 - 3x + 1

 

 

Graph the polynomial functions.

 

1. f(x) = x4 - 2x3 + x - 2  

2. f(x) = x3 + 2x2 - 3x + 1

 

 

 

 

Link to the polynomial function zero video:

 

http://videos.howstuffworks.com/hsw/11298-polynomial-functions-the-zero-property-video.htm

 

 

 

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