# Find all possible rational zeros of the equation

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Use the rational zeros theorem to list all possible rational zeros of the following

h(x)= -3 + 6x - 4x^3 + x^2 + x^4

Be sure that no value in the list appears more than once

Oct 16th, 2015

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The Rational Zeros Theorem states:

If P(x) is a polynomial with integer coefficients and if   is a zero of P(x) ( i.e. P() = 0 ), then p is a factor of the constant term of P(x) and q is a factor of the leading coefficient of P(x) .

First arrange the polynomial in descending order.

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

Constant term is -3.

Its factors are ±1 , ±3.

Coefficient of the leading term is 1

Its factors are ±1.

Possible solutions are $\pm\frac{1}{1},\pm\frac{3}{1}$ .

These can be simplified to ±1 , ±3.

So there are 4 possible solutions which are 1, -1, 3, -3.

Check whether each of them is a zero of h(x) by substituting for x in h(x) such that h(x) = 0.

For x = 1

$h(1)= 1^4-4\times1^3+1^2+6\times1-3=1-4+1+6-3=1\neq0$

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Oct 16th, 2015

In the same way checking for other values of x i.e. x = -1, x = 3 and x = -3

we get  $h(x)\neq0$

Hence, h(x) does not have any rational zeros.

Oct 16th, 2015

Oct 16th, 2015

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Oct 16th, 2015
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Oct 16th, 2015
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