To determine which of the given expressions is a factor of the polynomial \( 2x^2 + 9x - 35 \), we can use polynomial long division for each option.
Long Division Steps:
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Polynomial Division of \( 2x^2 + 9x - 35 \) by \( 2x - 5 \):
- First, divide \( 2x^2 \) by \( 2x \) to get \( x \).
- Multiply \( x \) by \( 2x - 5 \): \[ x(2x - 5) = 2x^2 - 5x \]
- Subtract this result from the original polynomial: \[ (2x^2 + 9x - 35) - (2x^2 - 5x) = 14x - 35 \]
- Now, divide \( 14x \) by \( 2x \) to get \( 7 \).
- Multiply \( 7 \) by \( 2x - 5 \): \[ 7(2x - 5) = 14x - 35 \]
- Subtract this from the previous remainder: \[ (14x - 35) - (14x - 35) = 0 \]
- Since we are left with a remainder of 0, \( 2x - 5 \) is a factor of \( 2x^2 + 9x - 35 \).
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Since \( 2x - 5 \) is a factor, we can conclude:
The answer is A. \( 2x - 5 \).
Optional Checks:
You could perform the same long division process for the other options, but since we have already established that \( 2x - 5 \) is a factor, we can confirm it without checking the remaining options, although it's fine to check them for further clarity.
Final Conclusion:
The factor of \( 2x^2 + 9x - 35 \) is: A. \( 2x - 5 \).
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