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

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

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

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

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

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



Answer :

To find the sum of the given polynomials and group like terms together, we'll follow these steps:

1. Write down the given polynomials:
[tex]\[ 10 x^2 y + 2 x y^2 - 4 x^2 - 4 x^2 y \][/tex]

2. Combine like terms:
- First, notice that \(10 x^2 y \) and \(-4 x^2 y \) are like terms because they both contain \( x^2 y \).
- Next, \(2 x y^2\) has no other like term.
- Finally, \(-4 x^2 \) has no other like term.

3. Combine the like terms:
- Combine \(10 x^2 y\) and \(-4 x^2 y\):
[tex]\[ 10 x^2 y - 4 x^2 y = 6 x^2 y \][/tex]
- The other terms \(2 x y^2\) and \(-4 x^2\) remain unchanged.

Therefore, the grouped polynomial is:
[tex]\[ 6 x^2 y + 2 x y^2 - 4 x^2 \][/tex]

Now, let's match this simplified polynomial with the provided choices:

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

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

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

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

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

From the choices, the correct expression that matches our simplified polynomial:
[tex]\[ 6 x^2 y + 2 x y^2 - 4 x^2 \][/tex]
is option 5.

Thus, the answer is:
[tex]\[ \left[10 x^2 y + 2 x y^2 + \left(-4 x^2 y \right) \right ] + \left(-4 x^2 \right) \][/tex]
This indicates the sum of the polynomials with like terms grouped together.