thanks in advance, Let (x) = a0 + a1x + a2x2 + + anxn epsilon Z[x], where an 0.

Question

thanks in advance,

Let (x) = a0 + a1x + a2x2 + + anxn epsilon Z[x], where an 0. In Pre-Calculus, the Rational Root Theorem states that if f(r/s ) = 0, then r|a0 and s|an. Consider f(x) = -5 + 3x + 2x3 epsilon Z[x]. What are all the factors (negative and positive) of -5? What are all the factors (negative and positive) of 2? From (a), list all the possible rational roots of f(x). Does f(x) have any rational roots? Prove Rational Root Theorem. Hints: Let r/s be in lowest terms such that f(r/s) = 0, where f(x) = a0 + a1x + a2x2 +...+anxn. How can you show that s|anrn and r|sna0? How can you use Euclid's Lemma afterwards?

Explanation / Answer

According to the rational zero theorem, any rational zero must have a factor of -5 in the numerator and a factor of 2 in the denominator.

p : factors of -5 ..... +1 , -1 , +5 , -5

q : factors of 2 ... +1 , -1 , +2 , -2

the possibilities of p/q in the simplest form are +1 ,-1 , +1/2 , -1/2 , +5 , -5 , +5/2 , -5/2

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