9.3. $\quad$ Find all quadratic trinomials $P(x)=x^{2}+b x+c$ such that $P(x)$ has integer roots, and the sum of its coefficients (i.e., $1+b+c$) is 10.
Answer: $(x-2)(x-11),(x-3)(x-6), x(x+9),(x+4)(x+1) . S o l u t i o n . ~ L e t ~ x_{1}<x_{2}$ be the roots. Then $P(x)=\left(x_{1}\right.$ $-x)\left(x_{2}-x\right)$ and $P(1)=10=\left(x_{1}-1\right)\left(x_{2}-1\right)$. From the factorizations of the number 10 into two factors, we obtain the possible values of the factors ( $x_{1}-1$ ) and ( $x_{2}-1$ ), namely: 1) 1 and $\left.10 ; 2\right) 2$ and $\left.5 ; 3\right)-10$ and $\left.-1,4\right)-5$ and -2 . Hence the result.