I need to determine the real coefficients $a, b, c$ of the following polynomial:
$$P(x) = x^5 + ax^4 - 2x^3 - 6x^2 + bx + c$$
I know that $P(-2) = 9$, and the sum of the solutions (roots of the polynomial) is 3, and so is the product. I'm not sure if these last two conditions are correct, they sound like a typo?
Then I need to write the given polynomial by degrees of $(x - 1)$.
Now, I've tried going this way:
If $P(-2) = 9$, then
$$16a - 2b + c = 37$$
As far as I know, a quintic polynomial should have 5 real solutions? Therefore,
$$x_1 \times x_2 \times x_3 \times x_4 \times x_5 = 3$$ and
$$x_1 + x_2 + x_3 + x_4 + x_5 = 3$$
But where do I go from here? How do I formulate this into something that will probably result in a system of two equations with $a, b, c$? Then I can solve them with the first one.
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