This would be a great place to apply synthetic division.
Is -1 a root of 5x^2 + 2x + 8 = 0? To find out, use -1 as divisor in synth. div."
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-1 / 5 2 8
-5 3
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5 -3 11 Since there's a remainder of 11, -1 is NOT a root of
5x^2 + 2x + 8 = 0. Move on to the next quadratic:
(x + 2)(x - 1)(x + 4). x=-1 is not a root. Move on to the next polynomial:
x^3 - 5x^2 + 2x + 8 = 0
Is -1 a root of that? Set up and carry out the synth. div.:
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-1 / 1 -5 2 8
-1 6 -8
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1 -6 8 0 This zero remainder tells us that -1 IS a root of
x^3 - 5x^2 + 2x + 8 = 0.
Move on to the next polynomial: x^4 - x^3 - 5x^2 + 2x + 8 = 0
Determine, using synth. div., whether -1 is or is not a root.
Once you've eliminated one or more of the polynomials,
You must, of course, determine whether or not 2 and 4 are roots.
The answer to this problem is the polynomial whose roots include -1, 2 and 4. This does not necessarily rule out the final choice, which has four roots.
