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If the function $-1 - 15 x - 35 x^2 - 28 x^3 - 9 x^4 - x^5 - 1/5 a (5 + 20 x + 21 x^2 + 8 x^3 + x^4)$ is plotted for different values of $a$, they all pass through several points that are independent from the value of $a$:

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What are these point and why do we see such a pattern? Can these points be used to formulate a general rule for roots of the main equation as a function of $a$?

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They are the roots of $5+20x+21x^2+8x^3+x^4$. These happen to be $\frac{-3 \pm \sqrt{5}}{2}$ and $\frac{-5\pm \sqrt{5}}{2}$.

EDIT: No, they don't have much to do with the roots of the main equation (which is a quintic, and in general won't have roots expressible by radicals).

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    Does that have anything to do with orthogonality of the two polynomials? In other words, is it correct to say that in general case, the roots of a linear combination of two polynomials doesn't have much two to do with roots of each of them separately? – 2017-01-19
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    If $x$ is a root of $Q$, then $x$ is a root of $P+aQ$ if and only if it is a root of $P$. If $x$ is not a root of $Q$, then $x$ is a root of $P + a Q$ for exactly one value of $a$, namely $-P(x)/Q(x)$. – 2017-01-19
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    Ok this is great. Thanks! I was wrong about another kind of connection based on some algebraic theorem that I might not know of. – 2017-01-19