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The degree of the polynomial determins how many roots it has.

If you have the linear polynomial $ax+b$ and the cubic polynomial $cx^3+dx^2+ex+f$, the highest power in the product $(cx^3+dx^2+ex+f)(ax+b)=acx^4+(ad+bc)x^3+(ae+bd)x^2+(af+be)x+bf$ is $x^4$ so the polynomial will have 4 roots.