Which statement is true about the polynomial 3x2y2 − 5xy2 − 3x2y2 2×2 after it has been fully simplified?
The correct Answer and Explanation is:
To simplify the polynomial (3x^2y^2 – 5xy^2 – 3x^2y^2 + 2x^2), let’s follow these steps:
Step 1: Combine Like Terms
We have two pairs of like terms that can be simplified. Like terms are terms that have the same variables raised to the same powers.
- The terms (3x^2y^2) and (-3x^2y^2) are like terms because they both involve (x^2y^2).
- The term (-5xy^2) does not have any like terms, so it remains unchanged.
- The term (2x^2) also does not have any like terms, so it remains unchanged.
Now, simplify:
[
3x^2y^2 – 3x^2y^2 = 0
]
So, the polynomial reduces to:
[
-5xy^2 + 2x^2
]
Step 2: Final Simplified Polynomial
After simplifying the like terms, the final polynomial is:
[
2x^2 – 5xy^2
]
Explanation of the Simplification Process
In this polynomial, we started by identifying like terms—terms that share both the same variables and the same exponents for those variables. The first and third terms, (3x^2y^2) and (-3x^2y^2), cancel each other out because their coefficients sum to zero. This leaves us with two terms that are not alike: (-5xy^2) and (2x^2). These terms cannot be combined because they involve different variables or different powers of those variables.
Thus, after combining like terms, the polynomial simplifies to (2x^2 – 5xy^2). This expression represents the simplest form of the original polynomial, as no further combination or factoring is possible.
Conclusion
The true statement about the polynomial (3x^2y^2 – 5xy^2 – 3x^2y^2 + 2x^2) is that, when fully simplified, it becomes (2x^2 – 5xy^2).