<s>[INST] Find $a$ and $b$ in the given cubic polynomial Find $a$ and $b$ such that $x+1$ and $x+2$ are factors of the polynomials $x^3+ax^2-bx+10$.
Here I am not sure that how can I obtain the value of $a$ and $b$, I tried to multiply $x+1$ and $x+2$ to obtain a quadratic equation by which I divided the obtained quadratic polynomial with the above given cubic polynomial, But it didn't worked. How can I overcome my answer.
 [/INST] If $x+1$ is a factor of $p(x) = x^3+ax^2-bx+10$, Then Sub. $x=-1$ in $p(x) = 0$
we Get $p(-1)=0\Leftrightarrow -1+a+b+10=0\Leftrightarrow a+b=-9$
Similarly If $x+2$ is a factor of $p(x) = x^3+ax^2-bx+10$, Then Sub. $x=-2$ in $p(x) = 0$
we Get $P(-2)=0\Leftrightarrow -8+4a+2b+10=0\Leftrightarrow 2a+b=-1$
After Solving these two equations, We Get $a=8$ and $b=-17$
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