Question

list the possible rational roots for x^6+5x^4+4x^2+3 as given by the rational roots theorem

Answer 1

Given polynomial x^6+5x^4+4x^2+3

By the rational root theorem

Since the leading coefficient is 1 and constant is 3.

So the possible zeroes are ±1,±3.

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

Given polynomial f (x ) = x ^6 + 5x ^4 + 4x ^2 + 3

If p /q  is a rational zero, then p  is a factor of 3 and q  is a factor of 1.

The possible values of p  are   ± 1 and ± 3.

The possible values for q  are ± 1.

So, p /q  = ± 1,  ± 3.

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

p /q	1	0	5	0	4	0	3
1	1	1	6	6	10	10	13
3	1	3	14	42	130	390	1173
-1	1	-1	6	-6	10	-10	13
-3	1	-3	14	-42	130	-390	1173

Since f (±1) is not equals to zero and f (± 3) is not equals to zero.

There is no rational zeros for x ^6 + 5x ^4 + 4x ^2 + 3 .