# Nature of roots of a quartic

I wonder if anyone can answer my question posted to the top-rated solution?

In case the link doesn't work, the given solution said:

"Let p(x)=x4−2x3−9x2+2x+8 and q(x)=ax+b. By sketching the graph of p, we can easily see that the only way that q can be tangent to p in two distinct points is if q is right at the bottom of p, touching each 2 down-humps in a different point, let us say, c and d.

Now let us sketch r(x)=p(x)−q(x). Because q is right at the bottom of p, the difference p(x)−q(x) is always positive, except for x=c and x=d, which make it go to zero. Therefore, r is a polynomial with only 2 real roots c and d. Furthermore, r is always positive when we get closer to c and to d, which means both have multiplicity 2 or greater."

and I am looking for explanation of the last sentence. Note by Matt McNabb
6 years, 8 months ago

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Staff - 6 years, 8 months ago

Thanks Calvin. I wasn't sure if it was or not.

- 6 years, 8 months ago