Polynomials?
Sketch the graph of a polynomial function that meets these requirements:

(1) Leading term is 2x^5
(2) Has two non-real complex roots
(3) Has a real root at x= -1/2
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answers:
Φ² = Φ+1 say: (3) Has a real root at x= -1/2 , so (2x+1) as a factor
(2) Has two non-real complex roots, so a simple (x²+1) as a factor will suffice.
(1) Leading term is 2x⁵, so we have the 2 covered at (1), and we have a leading term so far of 2x³. I will choose (x²-1) as the final factor. (Yes, you could factorize that further, but there is no need)

p(x) = (2x+1)(x²+1)(x²-1) = (2x+1)(x⁴-1) = 2x⁵+x⁴-2x-1

If you are graphing by hand, this polynomial passes through (-1,0), (-1/2,0), (0,-1), and (1,0).

Graph: https://www.desmos.com/calculator/86agpa...
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rotchm say: (1) implies 2(x-a)(x-b)(x-c)(x-d)(x-e). [why?].
(2) Choosing to use i & thus -i as the roots [why?] , we then have 2(x-i)(x+i)(x-c)(x-d)(x-e)

this equals 2(x²+1)(x-c)(x-d)(x-e)

(3) implies 2(x²+1)(x - -1/2) (x-d)(x-e) = 2(x²+1)(x +1/2) (x-d)(x-e).

You are now free to chose d & e. I would take d=e=0 for simplicity.

Thats your/a poly that satisfies the requirements. Expand it if need be. Done!
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Jeff Aaron say: https://www.wolframalpha.com/input/?i=Gr...
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