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I have to tackle a question related with roots of polynomial function. The graph of the polynomial is shown below:

enter image description here

(A)explain why, of the four roots of the equation f(x)=o, two are real and two are complex

I found I am a little bit messy with the concepts.....when f(x)=0, the two real roots are 2 and -4, how can I find the complex roots from the graph??

(B)The curve passes though the point (-1,-18). Find f(x) in the form (x)=(x-a)(x-b)(x^2+cx+d), were a, b, c, d belongs to Z

This question is simple, I put in 2,-4, and -32 to be a,b,c, and then plug in (-1,-18), and found that c=-33

(C)Find the two complex roots of the equation f(x)=0 in Cartesian form (D)express each of the four roots of the equation in the Euler’s form

Again, I know Cartesian form and euler’s form,but how can I find the complex roots?

Hope someone can help me ! Thanks!!

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  • $\begingroup$ The complex roots come from that quadratic, so apply the quadratic formula. $\endgroup$ – Randall Feb 26 at 4:16
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How can I find the complex roots of from the graph?

There's no easy answer to this question that I'm aware of. But fortunately that's not what you were asked. You were just asked to show that two are real and two are complex. You see the real ones with your eyes, and the other two must be complex. The fundamental theorem of algebra says there must be four, up to multiplicity, and since complex roots come in conjugate pairs, you know they are distinct, so there are two different ones.

I found that c = -33

That's not what I found. Also you didn't mention $d.$

How can I find the complex roots?

Once you have found $c$ and $d$, the complex roots will just be the roots of $x^2+cx+d,$ since the two real roots are already factored out. You can find the roots with the quadratic formula.

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  • $\begingroup$ For the 2nd question, I use 2 as a, -4 as b, and -32 as d, then plug in (-1,-18) and receives c=-33, can you tell me where the problem is? $\endgroup$ – 何雨桓 Feb 26 at 4:27
  • $\begingroup$ @Gnumbertester thanks $\endgroup$ – spaceisdarkgreen Feb 26 at 4:29
  • $\begingroup$ @何雨桓 d is not -32. $\endgroup$ – spaceisdarkgreen Feb 26 at 4:29
  • $\begingroup$ Thanks but how can I get d??? $\endgroup$ – 何雨桓 Feb 26 at 4:33
  • $\begingroup$ @何雨桓 You know (0,-32) is on the graph $\endgroup$ – spaceisdarkgreen Feb 26 at 4:34

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