Root within an interval

Question

Show the following: a. $p(x) =x^5−2x^4−3x^3+ 4x+ 1$ has a root in the interval $(1,2)$.

b. $f(x) =x^4+ 3x^3−5x^2−6$ has a root in the interval $(1,2)$.

Answer 1

for $p(x) =x^5−2x^4−3x^3+ 4x+ 1$ you have $p(1)=1>0$ and $p(2)=-15<0$, obviously polynomial is a continuous function, so using https://en.wikipedia.org/wiki/Intermediate_value_theorem we prove that there is $u\in(1,2)$ such that $p(u)=0$.

the second one is analogical.