SOLUTION: Please help me with solving this question below. It is in my algebra review test: Find the missing polynomial: (4x^2 - 3x + 2) - (x^2 + x + 2) + (?) = 5x^2 + 3x -2 so you

Question 28586: Please help me with solving this question below. It is in my algebra review test:
Find the missing polynomial:
(4x^2 - 3x + 2) - (x^2 + x + 2) + (?) = 5x^2 + 3x -2
so you would take the brackets away, but according to the distributive property, you would have to reverse the signs on the variables in the second brackets. thus:
4x^2 - 3x + 2 - x^2 -x -2 + (?) = 5x^2 + 3x - 2
then you group the like variables together right?:
3x^2 + 4x + (?) = 5x^2 + 3x -2
now i'm stuck, because i dont know what ONE variable will make the statement true. The answers in the back of my book say that the answer to this question is 2x^2 + 7x -2, which confuses me, because i thought that the question was asking for one missing variable... Am i reading the question wrong or am i not doing the question right?
Answer by sdmmadam@yahoo.com(530) (Show Source): You can put this solution on YOUR website!
Find the missing polynomial:
(4x^2 - 3x + 2) - (x^2 + x + 2) + (?) = 5x^2 + 3x -2
so you would take the brackets away, but according to the distributive property, you would have to reverse the signs on the variables in the second brackets. thus:
4x^2 - 3x + 2 - x^2 -x -2 + (?) = 5x^2 + 3x - 2
then you group the like variables together right?:
3x^2 + 4x + (?) = 5x^2 + 3x -2
now i'm stuck, because i dont know what ONE variable will make the statement true. The answers in the back of my book say that the answer to this question is 2x^2 + 7x -2, which confuses me, because i thought that the question was asking for one missing variable... Am i reading the question wrong or am i not doing the question right?

How much to present in the test?
Let the missing polynomial be [(A)x^2 +B(x)+C]
Then the given data is
(4x^2 - 3x + 2) - (x^2 + x + 2) + [(A)x^2 +B(x)+C]= 5x^2 + 3x -2 ----(*)
This implies (3+A)x^2+(B-4)x+C= (5)x^2+(3)x+(-2)
which further implies 3 +A = 5 giving A = 2
B-4 = 3 giving B = 7 and C= -2
Therefore the missing polynomial [(A)x^2 +B(x)+C] is
(2x^2+7x-2)
Note: While grouping the terms care should be exercised regarding the signs. for instance (-3x) added to (-x) is -4x and not +4x
The required answer is a polynomial which means the required answer is an expression in the one variable x in which the other expressions are in the problem.
One slight mistake that you have managed to make is while grouping the like terms. (muddle with the sign)That is because unless the rule is properly inculcated in the mind, we definitely ought to go through all the seemingly trivial but important detailed steps.
And the other thing is that there is a confusion in the mind regarding one polynomial (in the one variable x)and one term in x. The polynomial expression in x may (can) contain more than one term in the variable x. For example it might consist of a term in x^2, a term in x and a constant like the other expressions in x in the problem
Let us now go to the problem proper :
(4x^2 - 3x + 2) - (x^2 + x + 2) + (?) = 5x^2 + 3x -2 ----(*)
Let us assume the missing polynomial to be [(A)x^2 +B(x)+C]
Then (*) becomes (4x^2 - 3x + 2) - (x^2 + x + 2) + [(A)x^2 +B(x)+C]
= 5x^2 + 3x -2 ----(*)
Grouping like terms (along with the appropriate signs)which is nothing but additive commutativity and additive associativiity
(4-1+A)x^2+(-3-1+B)x+(2-2+C)= 5x^2 + 3x -2 ----(*)
(3+A)x^2+(B-4)x+C= (5)x^2+(3)x+(-2)
Equating coefficients of like terms
3 +A = 5 which gives A = 2
B-4 = 3 which gives B = 7
and C= -2
Therefore the missing polynomial [(A)x^2 +B(x)+C] is
(2x^2+7x-2)
How much to present in the test?
Let the missing polynomial be [(A)x^2 +B(x)+C]
Then the given data is
(4x^2 - 3x + 2) - (x^2 + x + 2) + [(A)x^2 +B(x)+C]= 5x^2 + 3x -2 ----(*)
This implies (3+A)x^2+(B-4)x+C= (5)x^2+(3)x+(-2)
which further implies 3 +A = 5 giving A = 2
B-4 = 3 giving B = 7 and C= -2
Therefore the missing polynomial [(A)x^2 +B(x)+C] is
(2x^2+7x-2)
The quickest way to do the problem
if steps are not required to be shown is to do the whole thing mentally.
How?
Inspect and understand that (4-1+something)x^2=5x^2
giving that something to be 2
(-3-1+something)x=3x gives that something =7
how?(the question is -4+?=3 giving ?=7
Similarly (2-2+?)=-2 gives this ? =-2
Therefore the missing polynomial is 2x^2+7x-2

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