<s>[INST] Sixth degree polynomial problem If the graph of $$y = x^6 - 10x^5 + 29x^4 - 4x^3 + ax^2$$  always lies above the line $y = bx + c$, except for $3$ points where the curve intersects the line.
What is the largest value of $x$ for which the line intersects the curve?


*

*A) 4

*B) 5

*C) 6

*D) 7


Through general idea of graphs and with the help of a graphing calculator I have concluded the approximate look of such a curve and a line.
However, I would like to know a proper mathematical solution to this problem.
 [/INST] As there are three double roots, $$x^6 - 10x^5 + 29x^4 - 4x^3 + ax^2-bx-c$$ is a perfect square, and we can evaluate its square root.
Looking at the first two terms,
$$x^6-10x^5\leftrightarrow(x^3-5x)^2=x^6-10x^5\cdots$$
Next
$$x^6-10x^5+29x^4\leftrightarrow(x^3-5x^2+px)^2=x^6-10x^5+(2p+25)x^4\cdots$$ so that $p=2$.
Then
$$x^6-10x^5+29x^4-4x^3\leftrightarrow(x^3-5x^2+2x+q)^2=x^6-10x^5+29x^4+(2q-20)x^3\cdots$$ and $q=8$.
The roots of 
$$x^3-5x^2+2x+8$$ are $-1, 2$ and $\color{green}4$.
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