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Which expression is equivalent to this polynomial expression?

[tex]\[ \left(5xy^2 + 3x^2 - 7\right) + \left(3x^2y^2 - xy^2 + 3y^2 + 4\right) \][/tex]

A. [tex]\[ 9x^2y^2 + 4xy^2 - 3 \][/tex]
B. [tex]\[ 3x^2y^2 + 6xy^2 + 6x^2 + 3 \][/tex]
C. [tex]\[ 3x^2y^2 + 4xy^2 + 3x^2 + 3y^2 - 3 \][/tex]
D. [tex]\[ 8x^2y^2 + 2xy^2 - 4y^2 + 4 \][/tex]

Sagot :

GinnyAnswer
To determine which expression is equivalent to the given polynomial expression:

[tex]\[ (5xy^2 + 3x^2 - 7) + (3x^2y^2 - xy^2 + 3y^2 + 4), \][/tex]

we will combine like terms step by step.

1. Collect terms involving [tex]\(x^2 y^2\)[/tex]:
- From the second polynomial: [tex]\(3x^2 y^2\)[/tex]
- Sum: [tex]\(3x^2 y^2\)[/tex]

2. Collect terms involving [tex]\(xy^2\)[/tex]:
- From the first polynomial: [tex]\(5xy^2\)[/tex]
- From the second polynomial: [tex]\(-xy^2\)[/tex]
- Sum: [tex]\(5xy^2 - xy^2 = 4xy^2\)[/tex]

3. Collect terms involving [tex]\(y^2\)[/tex]:
- From the second polynomial: [tex]\(3y^2\)[/tex]
- Sum: [tex]\(3y^2\)[/tex]

4. Collect terms involving [tex]\(x^2\)[/tex]:
- From the first polynomial: [tex]\(3x^2\)[/tex]
- Sum: [tex]\(3x^2\)[/tex]

5. Collect constants:
- From the first polynomial: [tex]\(-7\)[/tex]
- From the second polynomial: [tex]\(4\)[/tex]
- Sum: [tex]\(-7 + 4 = -3\)[/tex]

By combining all these, the equivalent polynomial expression is:

[tex]\[ 3x^2y^2 + 4xy^2 + 3y^2 + 3x^2 - 3 \][/tex]

Thus, the correct answer is:
C. [tex]\(3 x^2 y^2 + 4 x y^2 + 3 x^2 + 3 y^2 - 3\)[/tex]
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