Solve!!!!!!!!!!!

Question

x^3-x^2-x-2>0?

Answer 1

The inequality is x³ -x² - x -2 > 0.

Write the polynomial as factor form .

x³ -x² - x -2

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.

Rational Root Theorem, if a rational number in simplest form p/q is a root of the polynomial equation a_nx ⁿ + a_n – 1x ^{n – 1} + ... + a₁x + a₀ = 0, then p is a factor of a₀ and q is a factor if a_n.

The function is x³ -x² - x -2.

If p/q is a rational zero, then p is a factor of - 2 and q is a factor of 1.

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

The possible values for q are ± 1.

By the Rational Roots Theorem, the only possible rational roots are, p/q = ± 1,± 2.

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

p/q	1	-1	-1	-2
1	1	0	-1	-3
2	1	1	1	0

Since the equation has a zero at x = 2 .

x³ -x² - x -2 = (x-2)(x² + x + 1)

x² + x + 1 

x = (-b±√(b² -4ac ))/2a 

x = (-1±√(1² -4(1)(1) ))/2(1)

 x = (-1± i √3)/2

x³ -x² - x -2 = (x-2)(x² + x + 1)

Imaginary roots are neglected in case of solving inequality .

The key numbers is x = 2 . So, the polynomial’s test interval is  (2, ∞).

In test interval, choose a representative x-value and evaluate the polynomial.

Test Interval  x-value                Polynomial Value                                  Conclusion

(2, ∞)              x = 3               (3)³ -(3)² - 3 -2 = 27 - 9 -3 -2 = 13 > 0          Positive

From this we can conclude that the inequality is satisfied on the open intervals  (2, ∞).

So, the solution set is  (2, ∞) and its graph is