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Which expression shows the sum of the polynomials with like terms grouped together?

A. \(10 x^2 y + 2 x y^2 - 4 x^2 - 4 x^2 y\)

B. \(\left[\left(-4 x^2\right) + \left(-4 x^2 y\right) + 10 x^2 y\right] + 2 x y^2\)

C. \(10 x^2 y + 2 x y^2 + \left[\left(-4 x^2\right) + \left(-4 x^2 y\right)\right]\)

D. \(\left(-4 x^2\right) + 2 x y^2 + \left[10 x^2 y + \left(-4 x^2 y\right)\right]\)

E. [tex]\(\left[10 x^2 y + 2 x y^2 + \left(-4 x^2 y\right)\right] + \left(-4 x^2\right)\)[/tex]



Answer :

GinnyAnswer
To find the sum of the polynomials \(10x^2y + 2xy^2 - 4x^2\) and \(-4x^2y\), we should group and combine the like terms from each polynomial step by step. Let's work through this process:

1. Write down the given polynomials:

[tex]\[ P_1 = 10x^2y + 2xy^2 - 4x^2 \][/tex]

[tex]\[ P_2 = -4x^2y \][/tex]

2. Group the like terms:

Notice that in \(P_1\) and \(P_2\), we have terms involving \(x^2y\), \(xy^2\), and \(x^2\).

3. Combine the like terms:

- The terms involving \(x^2y\):
[tex]\[ 10x^2y + (-4x^2y) = (10 - 4)x^2y = 6x^2y \][/tex]

- The terms involving \(xy^2\):
[tex]\[ 2xy^2 \][/tex]
(since there's only one term involving \(xy^2\), it remains unchanged)

- The terms involving \(x^2\):
[tex]\[ -4x^2 \][/tex]
(again, since it’s the only term involving \(x^2\), it remains the same)

4. Write the sum of the polynomials by combining all the terms:

[tex]\[ 6x^2y + 2xy^2 - 4x^2 \][/tex]

Therefore, the expression that shows the sum of the polynomials with like terms grouped together is:

[tex]\[ 6x^2y + 2xy^2 - 4x^2 \][/tex]

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