Please help! how do you solve this equation?

Question

 Z^4 - 5*Z^2 + 10 * Z - 6 = 0?  

 

 

Answer 1

The equation z ⁴- 5z ² + 10z - 6 = 0

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 6 and q  is a factor of 1.

The possible values of p  are   ± 1,  ± 2, ± 3 and ± 6.

The possible values for q are ± 1.

By the Rational Roots Theorem, the only possible rational roots are, p/q = ± 1,  ± 2, ± 3 and ± 6.

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

p/q	1	0	-5	10	-6
-1	1	-1	-4	14	-20
1	1	1	-4	6	0

Since f (1) = 0,  z = 1 is a zero. The depressed polynomial is  z ³ + z ² - 4z + 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 and ± 6.

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,  z = - 3 is a zero. The depressed polynomial is  z ² - 2z + 2 = 0

Since the depressed polynomial of this zero, z ²- 2z + 2, is quadratic, use the Quadratic Formula to find the roots of the related quadratic equation

z ² - 2z + 2 = 0

Compare it to .

Roots are

The equation z ⁴- 5z ² + 10z - 6 = 0 have two real solutions at z = 1, - 3 and two imaginary solutions at z = 1 + 2i and 1 - 2i.