How do you use the rational roots theorem to find all possible zeros of $P (x) = 2 x^{5} + 3 x^{4} + 2 x^{2} - 2$ $P(x) = 2x^5 + 3x^4 + 2x^2 - 2$ ?

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How do you use the rational roots theorem to find all possible zeros of $P (x) = 2 x^{5} + 3 x^{4} + 2 x^{2} - 2$ $P(x) = 2x^5 + 3x^4 + 2x^2 - 2$ ?

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Answer 1 · 2016-04-25T22:51:17

We seek for rational roots within numbers $\frac{p}{q}$ $p/q$ , where $p ∣ a_{0}$ $p | a_0$ and $q ∣ a_{n}$ $q | a_n$ .
If $p ∣ - 2$ $p | -2$ then $p \in {\pm 1, \pm 2}$ $p in {+-1,+-2}$ .
If $q ∣ 2$ $q | 2$ then $q \in {\pm 1, \pm 2}$ $q in {+-1,+-2}$ .
Thus $\frac{p}{q} \in R = {\pm 1, \pm \frac{1}{2}, \pm 2}$ $p/q in R={+-1,+-1/2,+-2}$ .

Now, if the polynomian $P (x)$ $P(x)$ has any rational roots, they all are in $R$ $R$ so we proceed by trial and error:

$P (1) = 5 \neq 0$ $P(1)=5 !=0$
$P (- 1) = 1 \neq 0$ $P(-1)=1 !=0$
$P (0.5) = - 1.25 \neq 0$ $P(0.5)=-1.25 !=0$
$P (- 0.5) = - 1.375 \neq 0$ $P(-0.5)=-1.375 !=0$
$P (2) = 118 \neq 0$ $P(2)=118 !=0$
$P (- 2) = - 10 \neq 0$ $P(-2)=-10 !=0$

From this we can conclude that the polynomial $P (x)$ $P(x)$ has no rational roots.

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How do you use the rational roots theorem to find all possible zeros of $P (x) = 2 x^{5} + 3 x^{4} + 2 x^{2} - 2$ $P(x) = 2x^5 + 3x^4 + 2x^2 - 2$ ?

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How do you use the rational roots theorem to find all possible zeros of P(x)=2x5+3x4+2x2−2 P(x) = 2x^5 + 3x^4 + 2x^2 - 2?

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How do you use the rational roots theorem to find all possible zeros of $P (x) = 2 x^{5} + 3 x^{4} + 2 x^{2} - 2$ $P(x) = 2x^5 + 3x^4 + 2x^2 - 2$ ?