The thing to think about here is the π3 orbital. The answer key tells you that the π3 orbital determines the character of the radical – the picture of 3 lobes, one at each end and one in the middle is consistent with the resonance picture showing the radical electron at either end or in the middle.
You can also extend that to the situation where the electron is gone – the positive charge can only exist where there used to be an electron. Also, if you add an electron the anion pair can still only exist in those three locations.
Now here’s the thing: any linear carbon molecule that has 5 carbons will have the same π1 through π5* that are diagrammed out. That is true for radicals and non radicals alike. For example, 1,3 pentadiene will also have those 5 MO’s. However, in 1,3 pentadiene there are only 4 pi electrons floating around, so we never get to the π3 orbital, and we don’t have to consider its character. Whereas here you have 5 (the radical) or 4 (but you’re short one hydrogen, so there’s a + floating around) or 6 (the anion).
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