Find all the real zeros of he polynomial

Question

Find all the real zeros of the polynomial

P(x)=x⁴+x³-13x²-x+12

its real zeros are

x1=

x2=

x3=

x4= 

with x1≤x2≤x3≤x4

Answer 1

P(x) = x^4 + x^3 - 13x^2 - x + 12

                         +1              +1      = 2 or 0 positive real zeros

P(-x) = x^4 - x^3 - 13x^2 + x + 12

                 +1                  +1          = 2 or 0 negative real zeros

 

Any rational zero will be a factor of 12: ±1, ±2, ±3, ±4, ±6, ±12

 

1 | 1   1 -13  -1   12

          1    2 -11 -12           

3 | 1   2 -11 -12 |  0 = P(1)

          3  15  12        

     1   5    4 | 0 = P(3)

 

P(x) =(x-1)(x-3)(x^2+5x+4)

P(x) =(x-1)(x-3)(x+4)(x+1)

 

Real zeros of P(x): x = -4, -1, 1, 3