Question

Solve F(x)=3X^3-8X^2+X+2?

Answer 1

The function f(x) = 3x³ - 8x² + x + 2

Zeros are solutions of the function.

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.

If p/q is a rational zero, then p is a factor of 2 and q is a factor of 3.

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

The possible values for q are ± 1 and ± 3.

So, p/q = ± 1, ± 2, ± 1/3 and ± 2/3.

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

p/q	3	- 8	1	2
1	3	- 5	- 4	- 2
2	3	- 2	- 3	- 4
1/3	3	- 7	- 17/3	1/9
2/3	3	- 6	- 3	0

Since f(2/3)  =  0,  x = 2/3 is a zero. The depressed polynomial is  3x² - 6x - 3 = 0

x² - 2x - 1 = 0

Use the Quadratic Formula to find the roots of the related quadratic equation x² - 2x - 1 = 0.

Roots are x = [- b ± √(b² - 4ac)]/2a

b² - 4ac = (- 2)² - 4(1)(- 1) = 4 + 4  = 8

x = [- (- 2) ± √8 ]/2(1)

x = [ 2 ± 2√2 ]/2

x = 2[ 1 ± √2 ]/2

x = (1 ± √2 )

x = 1 + √2 and 1 - √2

Solutions of f(x) = 3x³ - 8x² + x + 2 are x = 2/3, 1 + √2 and 1 - √2.