The smallest integer that can be added to -2m^3 − m + m^2 + 1 to make it completely divisible by m + 1 is:
Well, well, well, looks like we have ourselves a math problem. Don't worry, I'm here to clown around and help you out!
Now, to find the smallest integer that can be added to make the expression completely divisible by m + 1, we need to perform some math magic.
Let's rewrite the expression: -2m^3 + m^2 - m + 1.
To make it divisible by m + 1, we divide the expression by m + 1 using good ol' long division (don't worry, I'll do the calculations for you).
-2m^2 + 3m - 4
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m + 1 │ -2m^3 + m^2 - m + 1
After dividing, we're left with a remainder of -4.
So, the smallest integer that can be added to the expression is -4!
And voila, now it's entirely divisible by m + 1. Math problems may be serious, but with a little humor, we can make them a bit more enjoyable!
To find the smallest integer that can be added to a given expression to make it completely divisible by another expression, we can use the concept of polynomial long division.
Let's perform the division of the given expression (-2m^3 − m + m^2 + 1) by (m + 1):
-2m^2 +3m -4
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m + 1 | -2m^3 + m^2 + m + 1
- (-2m^3 - 2m^2)
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- m^2 + m + 1
- (-m^2 - m)
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2m + 1
- (2m + 2)
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- 1
The result of the division is -2m^2 + 3m - 4 with a remainder of -1.
To make the expression completely divisible by m + 1, we need to add the negative of the remainder (-1) to the given expression:
-2m^3 − m + m^2 + 1 - 1 = -2m^3 − m + m^2 + 0
Therefore, the smallest integer that can be added to -2m^3 − m + m^2 + 1 to make it completely divisible by m + 1 is 0.
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.
interpretation:
the division must be excact, that is,
there cannot be a remainder.
Arrange the terms in the proper order and do a synthetic division:
-1 | -2 1 -1 1
....... 2..-3..4
.... -2 3..-4..5
so we should have had a 0 instead of a 5 at the end of our algorithm, that means the constant at the end of our function should have been -4
But we had a 1, so adding -5 would have done the trick