To demonstrate polynomial subtraction correctly, we start by handling the expression piece by piece.
The expression we need to evaluate is: \[ (2x^2 + 6x + 1) - (3x^2 + 3x + 9) \]
Step 1: Distribute the negative sign across the second polynomial: \[ = 2x^2 + 6x + 1 - 3x^2 - 3x - 9 \]
Step 2: Combine like terms:
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Combine \(2x^2\) and \(-3x^2\): \[ 2x^2 - 3x^2 = -x^2 \]
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Combine \(6x\) and \(-3x\): \[ 6x - 3x = 3x \]
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Combine \(1\) and \(-9\): \[ 1 - 9 = -8 \]
Putting it all together: \[ -x^2 + 3x - 8 \]
Now, let's evaluate the provided options to find the correct demonstration of polynomial subtraction.
- First option: \((2x^2 + 6x + 1) - (3x^2 + 3x + 9) = -x^2 + 3x - 8\) (Correct)
- Second option: \((2x^2 + 6x + 1) - (3x^2 + 3x + 9) = x^2 + 3x + 8\) (Incorrect)
- Third option: \((2x^2 + 6x + 1) - (3x^2 + 3x + 9) = 5x^2 + 9x + 10\) (Incorrect)
- Fourth option: \((2x^2 + 6x + 1) - (3x^2 + 3x + 9) = -5x^2 + 3x - 8\) (Incorrect)
The first option is the correct example of polynomial subtraction.