how to solve P(x)=x^3-5x^2+2x+8

Question

find all rational roots for P(x)=0.

Answer 1

Identify Rational Zeros :

Usually it is not practical to test all possible zeros of a polynomial function using only synthetic substitution. The Rational Zero Theorem can be used for finding the some possible zeros to test.

P(x) =  x³ - 5x² + 2x + 8 = 0

If p/q is a rational zero, then p is a factor of 8 and q is a factor of 1.

The possible values of p are   ± 1,   ± 2, and   ± 4.

The possible values for q are ± 1.

So, p/q = ± 1,   ± 2, ± 4.

Make a table for the synthetic division and test possible  zeros.

p/q	1	- 5	2	8
1	1	- 4	- 2	6
2	1	- 3	- 4	0
4	1	- 1	- 2	0
-1	1	- 6	8	0
-2	1	- 7	16	- 24

Since P(-1)   =   0, x   =   -1 is a zero. The depressed polynomial is   x² - 6x +8 = 0.

Since the depressed polynomial of this zero, x² - 6x +8, is quadratic, use the Quadratic Formula to find the roots of the related quadratic equation

x = [- b ± sqrt(b^2 - 4ac)]/2a

Substitute b = -6, a  = 1, and c = 8.

x = [6 ± sqrt(36 - 32)]/2

x = [6 ± sqrt(4)]/2

x = [6 ± 2]/2

x = (6 + 2)/2, (6 - 2)/2

x = 8/2, 4/2

x = 4, 2.

The function has one real zero at x = -1, x = 2 and x = 4.