find all the real zeros of the polynomial

Question

find all the real zeros of the polynomial 

P(x)=x³-2x²-9x+4

The real zeros of P(x) have x=

Answer 1

P(x) = x^3 - 2x^2 - 9x + 4

 

If there are rational zeros, p/q, then

p is a factor of 4 and q is a factor of 1.

 

So a rational zero would have to be one of:

 

±1, ±2, ±4

 

Using mental math ±1 don't work.

 

2 | 1 -2 -9   4

         2  0 -18

     1  0 -9 | not zero

 

-2 | 1 -2 -9  4
         -2  8  2   
      1 -4 -1 | not zero

 

4 | 1 -2 -9    4

         4  8   -4

     1  2 -1 | 0

 

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

 

0 = x^2 + 2x - 1

0 = x^2 + 2(1)x +1^2 -1^2 - 1

0 = (x + 1)^2 - 2

(x + 1)^2 = 2

|x + 1| = √(2)

x + 1 = ±√(2)

x = -1 ± √(2)

 

So the real zeros of P(x) are x = -1 - √(2), -1 + √(2), and 4.