I am trying to find the rational zeros of 2x^5 + 3x^3 + 4x^2 − 10 , but I am stuck at factoring the equation? Would somebody please explain?

There are none.

There is at most 1 positive real root, according to Descartes' rule of signs. It is near 1, but not 1, thus not rational.

The roots are approximately
.. 1.03546
.. -1.02267 ± 0.729744 i
.. 0.504936 ± 1.67463 i
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A graphing calculator can be really helpful when trying to factor higher degree polynomials.

find the factors, including 1, for each of the 'end' coefficients. [the highest x element and the constant]. assume that the form you are looking for is (ax^n+c)(bx^m+d)

it follows that ab = 2, cd = -10 and the combinations ad and cb must result in the coefficients for the intermediate terms [that's three and four]

you now have four equations in four unknowns, which can be solved if there are any unique solutions.

oh, yes ... n + m = 5 where one is 2 and the other 3 [from the exponents on the x elements]


ps -- this also works with fractions as the a, b, c, and d

Look up the rational root theorem. It will show you how to create a list of possible roots.

p = factors of 10
q = factors of 2
r = +/- p/q
r = +/- {1 2 5 10} / {1 2}
+/- {1/2 1 2 5/2 5 10}
none of these satisfy f(x)=0

There is one real root ~ 1.03.