To fill in the blanks for the polynomial equation that shows the subtraction of the second expression from the first, we can proceed step-by-step with the given expressions.
The first expression is: \( 9x^2 + x + 13 \)
The second expression to subtract is: \( 2x^2 + 12x + __ \)
The third expression is: \( x^2 + 5x + 2 \)
We need to subtract the second and third expressions from the first expression:
- Combine the second and third expressions for subtraction. Let's first evaluate:
\[ 2x^2 + 12x + (x^2 + 5x + 2) = 2x^2 + 12x + x^2 + 5x + 2 = (2x^2 + x^2) + (12x + 5x) + 2 \]
This results in: \[ 3x^2 + 17x + 2 \]
- Now we perform the subtraction from the first expression:
\[ 9x^2 + x + 13 - (3x^2 + 17x + 2) \]
Distributing the negative sign, we have: \[ 9x^2 + x + 13 - 3x^2 - 17x - 2 \]
- Now combine like terms:
- For \(x^2\): \(9x^2 - 3x^2 = 6x^2\)
- For \(x\): \(x - 17x = -16x\)
- For the constant term: \(13 - 2 = 11\)
The resulting difference is: \[ 6x^2 - 16x + 11 \]
- So, the blanks can be filled with:
- The second expression gives us a missing constant term: \(2\)
- The resulting polynomial from the subtraction is: \(6x^2 - 16x + 11\)
Thus, putting it all together, the complete polynomial equation can be represented as:
\[ 9x^2 + x + 13 - (2x^2 + 12x + 2) = 6x^2 - 16x + 11 \]
Completion of the statement:
- The blank for the second expression is 2, and based on the subtraction and combination, the resulting difference is 6x² - 16x + 11.
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