To find the smallest integer that can be added to an expression to make it completely divisible by another expression, we can use the concept of remainder theorem.
In this case, we want to make the expression -2m^3 − m + m^2 + 1 completely divisible by m + 1. We can rewrite the expression as:
-2m^3 + m^2 - m + 1
Using the remainder theorem, we know that if this expression is completely divisible by m + 1, then the remainder when we divide this expression by m + 1 should be zero.
Let's perform the polynomial division to find the remainder:
-2m^2 - 3m + 4
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m + 1 | -2m^3 + m^2 - m + 1
To perform the long division, we start by dividing -2m^3 (the term with the highest power) by m. This gives us -2m^2, which we write on top. Then we multiply m + 1 by -2m^2 to get -2m^3 - 2m^2 and subtract it from the original expression. We repeat this process until we have divided all the terms.
-2m^2 - 3m + 4
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m + 1 | -2m^3 + m^2 - m + 1
- (-2m^3 + 2m^2) = 3m^2 - m
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- 4m + 1
- (-4m + 4) = -3
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1
The remainder is 1, which implies that -2m^3 + m^2 - m + 1 divided by m + 1 results in a remainder of 1, not zero.
Therefore, to make -2m^3 − m + m^2 + 1 completely divisible by m + 1, we need to add the value of the remainder, which is 1, to the expression.
So, the smallest integer that can be added to -2m^3 − m + m^2 + 1 to make it completely divisible by m + 1 is 1.