SOLUTION: Find all solutions of the equation. (Enter all answers including repetitions. Enter your answers as a comma-separated list.) x^4 + 3x^3 − 13x^2 − 9x + 30 = 0

Algebra ->  College  -> Linear Algebra -> SOLUTION: Find all solutions of the equation. (Enter all answers including repetitions. Enter your answers as a comma-separated list.) x^4 + 3x^3 − 13x^2 − 9x + 30 = 0      Log On


   

Question 1147054: Find all solutions of the equation. (Enter all answers including repetitions. Enter your answers as a comma-separated list.)
x^4 + 3x^3 − 13x^2 − 9x + 30 = 0
Found 3 solutions by Alan3354, ikleyn, Edwin McCravy:
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
(Enter all answers including repetitions. Enter your answers as a comma-separated list.)
===============
Why do I have to do that?'

Answer by ikleyn(52847) About Me  (Show Source):
You can put this solution on YOUR website!
.

They want you apply the  Rational  Roots  theorem.


But I prefer to make a plot of the polynomial first and then to see its roots VISUALLY.


When you see them visually, it is  MUCH  EASIER  to apply then the  Rational  Roots  theorem

and to justify everything, pretending as if you  "guessed"  these roots  using yours  6-th sense".



Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!

x%5E4+%2B+3x%5E3+-+13x%5E2+-+9x+%2B+30+=+0

The potential rational solutions are the positive and negative fractions whose
numerator is a factor of the constant term, 30, and whose denominator is a
factor of the coefficient of the largest power of x, which is 1.

Potential rational solutions: 

Plot it on your TI-83 or 84:

Press WINDOW and make it read:

WINDOW
 Xmin=-10
 Xmax=10
 Xscl=1
 Ymin=-100
 Ymax=100
 Yscl=1

Press Y= and make it read:
\Y1=X^4+3X^3-13X^2-9X+30

Press GRAPH

You get something like this:

graph%28400%2C300%2C-10%2C10%2C-100%2C100%2Cx%5E4%2B3x%5E3-13x%5E2-9x%2B30%29

The rational solutions are the x-coordinates of the x-intercepts.

You see that there is an x-intercept at -5, so we use synthetic

division:

-5 | 1    3  -13   -9   30
   |     -5   10   15  -30
     1   -2   -3    6    0

This factors the left side as

%28x%2B5%29%28x%5E3-2x%5E2-3x%2B6%29=0

It appears to have a root at 2:

 2 | 1   -2   -3    6   
   |      2    0   -6      
     1    0   -3    0

This further factors as

x%2B5%29%28x-2%29%28x%5E2-3%29=0

We set each factor on the left = 0:

x+5=0;    x-2=0;    x²-3=0
  x=-5      x=2;      x²=3
                       x=±√3

The four solutions are -5, 2, √3, -√3

Edwin