Krdytkyu

My question: When the expression $x^3+ax^2-x+2$ is divided by $x+1$, the remainder is four times as great as the remainder when the expression is divided by$x-2$. Find the value of $a$.

My problems: I couldn't quite get the pieces of information that are provided, for instance 'the remainder is four times as great as the remainder' part.

Can someone please show me some working outs? Thank you!

The question essentially says that $$f(-1) = 4cdot f(2)$$ On simplifying we get, $$2+a = 8+4a$$ Giving us $$a=-2$$

When we divide some polynomial(dividend) by another polynomial(divisor), we get a quotient polynomial, and a remainder polynomial, whose degree is certainly less than the degree of the divisor polynomial.

The question is telling us the relationship between two remainders : one that you get when you divide $x^3+ax^2-x+2$ by $x+1$, and the other that you get when you divide $x^3+ax^2 - x+2$ by $x-2$. It is then asking you to find the value of $a$ with this information. In more simple terms :

Let $r_1$ be the remainder obtained when the polynomial $x^3+ax^2 - x +2$ is divided by the polynomial $x+1$. Let $r_2$ be the remainder obtained when the polynomial $x^3 + ax^2 - x + 2$ is divided by $x-2$. It is known that $r_1 = 4 times r_2$. Find the value of $a$.

If this rewrite is also unclear, please inform me.

EDIT : The remainder theorem is the key to finding $r_1$ and $r_2$ above, since it expresses these quantities in terms of the dividend and divisor polynomial.
Knowing that $r_1 = 4r_2$ then leaves us with an linear equation where $a$ is the sole variable.