possible rational roots for function f(x)=x^4-2x^3-7x^2+18x-18

Question

all the possible rational roots .

Answer 1

The function F ( x ) = x⁴ - 2x³ - 7x² + 18x - 18.

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_nxⁿ + a_n – 1x^{n – 1} + ... + a₁x + a₀ = 0, then p is a factor of a₀ and q is a factor if a_n.

If p/q is a rational zero, then p is a factor of 18 and q is a factor of 1.

The possible values of p are   ± 1, ± 2, ± 3.

The possible values for q are ± 1.

By the Rational Roots Theorem, the only possible rational roots are, p/q = ± 1, ± 2, ± 3.

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

p/q	1	- 2	- 7	18	- 18
1	1	- 1	- 8	10	- 8
- 1	1	- 3	4	14	- 32
3	1	1	- 4	6	0

Since f(3) = 0, x = 3 is a zero. The depressed polynomial is  x³ + x² – 4x + 6 = 0.

Answer 2

Contd.....

If p/q is a rational zero, then p is a factor of 6 and q is a factor of 1.

By the Rational Roots Theorem, the only possible rational roots are, p/q = ± 1, ± 2, ± 3.

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

p/q	1	1	- 4	6
1	1	2	- 2	4
- 1	1	0	- 4	10
- 3	1	- 2	2	0

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

Since the depressed polynomial of this zero, x² – 2x + 2, is quadratic, use the Quadratic Formula to find the roots of the related quadratic equation x² – 2x + 2.

x = [-b ± √(b^2 - 4ac)]/2a

Substitute b = - 2, a = 1, and c = 2.

x = [-(- 2) ± √((- 2)^2 - 4 * 1 * (2))]/2 * 1

x = [ 2 ± √(4 - 8)]/2

x = [ 2 ± √- 4]/2

Substitute the value - 1 = i ^2.

x = [ 2 ± 2√(i ^2)]/2

x = 1 ± i.

The function has two real zeros at x = 3, - 3 and two imaginary zeros at x = 1 + i and x = 1 - i.