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Oct 1, 2018

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a) Yes, as long as a and b are allowed to be complex numbers.  Consider the simple parabola

$$y = x^2+1 \\ \text{with complex roots } \pm i \\ \text{this can be written as }\\ (x-i)(x+i) = x^2 +i x - i x +(i)(-i) = x^2+1$$

b) No.  Consider a general cubic polynomial

$$y = a x^3 + b x^2 + c x + d$$

$$\text{as }x \to -\infty, \text{the }x^3 \text{ term dominates everything and }y \to -\infty \\ \text{as }x \to +\infty, \text{ likewise the }x^3 \text{ term dominates and }y \to +\infty \\ \text{somewhere along the way it's got to cross the x axis and at that point }\\ y=0 \text{ at a real value of x}$$

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Oct 1, 2018