Apply Descartes' Rule of Signs to f(x) = x to the fourth - 3x cubed + 2x squared + x - 6. How many negative real zeros can it have?

1. Negative real zeros are counted from the sign changes in f of negative x. 2. Replace x by negative x term by term, keeping careful track of the powers. 3. The fourth power keeps its plus sign, and the cubic term flips to plus 3x cubed. 4. The squared term stays plus 2x squared, the linear term flips to negative x, and the constant stays negative 6. 5. So the signs read plus, plus, plus, minus, minus. 6. There is exactly one place where the sign changes, from plus to minus. 7. The count of negative real zeros equals that number of sign changes, or falls short of it by an even number. 8. One minus two is negative, so no smaller count is allowed, and the answer is exactly one. 9. Exactly three would use the sign changes of f(x) itself, which govern positive zeros instead. 10. Exactly two and exactly four would each need more sign changes than f of negative x actually shows. _Source: OpenStax Precalculus (CC BY 4.0), section 3.6 Zeros of Polynomial Functions_