<s>[INST] If $a - b = 3$, $a + b + x = 2$, then find the value of $(a - b)[x^3 - 2ax^2 + a^2x - (a + b)b^2]$ After solving the question I got to the conclusion that it has something to do with $(a-b)$ being a factor of the whole polynomial. And since the value of $(a-b)$ is already given we can further solve the equations to get the value of $a$,$b$ and $x$ and substitute the values obtained to find the answer. But I am stuck on finding the value of x using the fact that $(a-b)$ is a factor.
If anyone knows the answer please share it. Thank you
 [/INST] Let $f(x)=(a-b)[x^3-2ax^2+a^2x-(a+b)b^2]$. Now using the Remainder Theorem, which tells us that if you divide a polynomial $f(x)$ by $(x-y)$ then the remainder is $f(y)$. Now, we know that $(a-b)$ is a factor of $f(x)$ so $x=3$ is a root. Substituting it in $a+b+x=2$ we get $$a+b+3=2$$ $$\implies a+b=-1$$ Now add $a-b=3$ and $a+b=-1$ getting $$a-b+a+b=3-1$$ $$\implies 2a=2$$ $$\implies a = 1$$. Now substitute the value of $a$ in $a-b=3$ we get $b=-2$. Now putting all these three values in $f(x)$, we get
$$f(x)=(1-(-2))[3^3-2\times 1 \times 3^2 + 1^2 \times 3 - (1-2)(-2)^2]$$
$$=(1+2)[27-18+3-(-4)]$$
$$=3(27-18+3+4)$$
$$=3(34-18)$$
$$=3 \times 16$$
$$=48$$
Hence the value of $f(x)$ is $48$
Hope it helps
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