How can you determine where two polynomials cross and are tangent to each other, just by factorising the difference?

Question

My textbook asked me to factorise the difference between P(x) and Q(x) and describe the intersection.
I'm having trouble understanding:

1. Why doing this would allow me to describe the intersection
2. And how to describe the intersection

Question from my textbook:
$P(x) = x^4 + 4x^3 − x + 5$ $Q(x) = x^3 − 3x^2 − 2x + 5$

Any help would be appreciated!

Answer 1

Points of intersection are when $P(x) = Q(x) \iff P(x) - Q(x) = 0$ this is, i think, the difference you are looking for.

Hope this helps.