Question

Use the rational zeroes theorem to state all the possible series of the following polynomial:

Answer 1

Given the function:

$f(x)=3x^6+4x^3-2x^2+4$

Let's use the rational zeros theorem to state all the possible zeros of the function.

Every zero of the function will have the form:

$\frac{p}{q}$

Where:

p is a factor of the last term(constant)

q is a factor of the leading coefficient.

Where:

Leading coefficient, q = 3

Factors of 3 = ±1, ±3

Constant term, p = 4

Factors of 4 = ±1, ±2, ±4

$\begin{gathered} \frac{p}{q}=±\frac{1}{1},\pm\frac{1}{3},\operatorname{\pm}\frac{2}{1},\operatorname{\pm}\frac{2}{3},\operatorname{\pm}\frac{4}{1},\operatorname{\pm}\frac{4}{3} \\ \\ Now\text{ simplify:} \\ \frac{p}{q}=\operatorname{\pm}1,\operatorname{\pm}\frac{1}{3},\operatorname{\pm}2,\operatorname{\pm}\frac{2}{3},\operatorname{\pm}4,\operatorname{\pm}\frac{4}{3} \end{gathered}$

Therefore, the possible zeros of the polynomial are:

$\pm1,\operatorname{\pm}\frac{1}{3},\operatorname{\pm}2,\operatorname{\pm}\frac{2}{3},\operatorname{\pm}4,\operatorname{\pm}\frac{4}{3}$

ANSWER:

$\pm1,\operatorname{\pm}\frac{1}{3},\operatorname{\pm}2,\operatorname{\pm}\frac{2}{3},\operatorname{\pm}4,\operatorname{\pm}\frac{4}{3}$