Precalculus help

Question

How do I find the possible rational zeros of the function g(x) = x^3 -5x^2 -7x +6

Answer 1

Since the leading coefficient is 1 and the constant coefficient is 6, possible rational zeroes are all factors of 6 (1, 2, 3, 6)divided by all factors of 1, which is of course just 1; the factors can be positive or negative, so all possible rational factors are ±1, ±2, ±3, ±6

It turned out that only 6 was a rational factor.

Using synthetic division

6 | 1 -5 -7 6

6 6 -6

1 1 -1

x^2 + x - 1, from the rational root theorem, would only have roots of 1 and -1

f(1) = -1

f(-1) = -1

Thus, we only have irrational factors.

It turns out that the irrational roots are (-1 ± √5)/2

Answer 2

Read the Rational Root Theorem!

The possible rational roots are ±1, ±2, ±3, and ±6