Find the bounds on the real zeros of the following function. f(x)= x^5-6x^4+9x^3-2x+7

Question

I need help solving this problem .  I am in college algebra.  Please help me.

Answer 1

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 ^5 - 6x ^4 + 9x ^3 - 2x  + 7

If p /q  is a rational zero, then p  is a factor of 7 and q  is a factor of 1.

The possible values of p  are   ± 1and  ± 7.

The possible values for q  are ± 1.

So, p /q  = ± 1,  ± 7.

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

p /q	1	-6	9	0	-2	7
1	1	-5	4	4	2	9
7	1	1	16	112	782	5481
-1	1	-7	16	-16	14	-7
-7	1	-13	100	-700	4898	34279

f (±1) is not equls to 0 and f (± 7) is not equals to zero.

There is no rational zeros for f (x ) = x ^5 - 6x ^4 + 9x ^3 - 2x + 7.