Confirming my answer

Question

So the question is... $P(x)$ is a polynomial of degree $2$.

When $P(x)$ is divided by $(x-1)$ and $(x-2)$ the remainders are $-6$ & $-5$ respectively

$(2x+1)$ is a factor of $P(x)$. Find polinomial $P(x)$....

And my answer is $P(x) = 2(x-1)(x-2) + (x-7) = 2x^2 - 5x - 3$... Is this the correct answer? And plus I did it by considering the remainders $-6$ & $-5$ and then used the factor $(2x + 1)$ to get the $2$...

If there is any other way to find the answer please let me know and if my answer is wrong the correct method will be helpful... thank u

Answer 1

$P(1)= -6$ and $P(2)=-5$ and $P(-1/2)=0$

Write $P(x)=ax^2+bx+c$ and solve the system:

\begin{eqnarray} -6 &=& a+b+c\\ -5 & = & 4a+2b+c\\ 0 &=& a/4-b/2+c \end{eqnarray}

Answer 2

Another way where you end with two unknown instead of three.

Put $P(x)=(2x+1)(ax+b)$ and use the fact that $P(1)=-6$ and $P(2)=-5$