To determine which example correctly demonstrates polynomial addition, we need to perform the addition of the polynomials \( (2x^2 + 6x + 1) \) and \( (3x^2 + 3x + 9) \) step by step:
- Combine like terms:
- \( (2x^2 + 3x^2) = 5x^2 \)
- \( (6x + 3x) = 9x \)
- \( (1 + 9) = 10 \)
So the correct addition of the two polynomials is:
\[ (2x^2 + 6x + 1) + (3x^2 + 3x + 9) = 5x^2 + 9x + 10 \]
Now, let's evaluate the options given:
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First option: \((2x^2 + 6x + 1) + (3x^2 + 3x + 9) = 5x^2 + 9x + 10\) - This is correct.
-
Second option: \((2x^2 + 6x + 1) + (3x^2 + 3x + 9) = 13x^2 + 9x + 10\) - Incorrect; \(2x^2 + 3x^2\) does not equal \(13x^2\).
-
Third option: \((2x^2 + 6x + 1) + (3x^2 + 3x + 9) = 5x^4 + 9x^2 + 10\) - Incorrect; combining the terms cannot result in an \(x^4\) term from these.
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Fourth option: \((2x^2 + 6x + 1) + (3x^2 + 3x + 9) = -x^2 + 3x - 8\) - Incorrect; this does not represent the correct addition at all.
Conclusion:
The correct choice is:
\((2x^2 + 6x + 1) + (3x^2 + 3x + 9) = 5x^2 + 9x + 10\) (first option).