a A polynomial with 7 real solutions has 7 roots and a lowest possible degree of 7. Assuming it has only linear factors, it must intercept the x-axis seven times.
B
b A graph can only have a combination of complex and real roots. Considering this, what must the polynomial's degree be?
C
c Assume that the graph only has complex solutions.
D
d A graph can only have a combination of complex and real roots. Considering this, what must the polynomial's degree be?
A
aExample Solution:
B
bExample Solution:
C
cExample Solution:
D
dExample Solution:
a If the polynomial only has linear factors, 7 real solutions reflect a polynomial that intercepts the x-axis 7 times. This means the lowest degree our function can have is 7. Assuming it is a seventh degree function with only linear factors, it will change direction six times. We see an example of this below.
b Note that a function can only have complex or real solutions. This means a combination of 5 real and 2 complex solutions must reflect a polynomial of the seventh degree, that intersects the x-axis five times. We see an example of this below.
c Assuming that the polynomial only has complex solutions, this must be a fourth degree polynomial that does not intersect the axis at all. We see an example of this below.
d Like in Part B, we recognize that a function can only have complex or real solutions. This means a combination of 4 real and 2 complex solutions must describe a polynomial of the sixth degree, and it intersects the x-axis four times. We see an example of this below.
