+0  
 
0
302
1
avatar+503 

1.

 

2.

 

 Oct 1, 2018
 #1
avatar+6186 
+1

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

 

\(y = a x^3 + b x^2 + c x + d\)

 

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

.
 Oct 1, 2018

11 Online Users