To complete the polynomial equation showing the subtraction of the second expression from the first, we need to perform the subtraction step by step.
Given expressions:
- \(9x^2 + x + 13\) (first expression)
- \(2x^2 + 12x + ( \ \ )\) (second expression, where we need to find the constant)
We rearrange the second expression with the last polynomial as \(x^2 + 5x + 2\).
To perform the subtraction, we subtract the entire second expression from the first: \[ (9x^2 + x + 13) - (2x^2 + 12x + (x^2 + 5x + 2)) \]
Let's combine the second expression: \[ 2x^2 + 12x + (x^2 + 5x + 2) = 2x^2 + 12x + x^2 + 5x + 2 \]
We can now combine like terms in the second expression:
- For \(x^2\): \(2x^2 + x^2 = 3x^2\)
- For \(x\): \(12x + 5x = 17x\)
- The constant is \(2\).
So, \[ 2x^2 + 12x + (x^2 + 5x + 2) = 3x^2 + 17x + 2 \]
Now let's subtract this combined expression from the first:
- For \(x^2\): \(9x^2 - 3x^2 = 6x^2\)
- For \(x\): \(x - 17x = -16x\)
- For the constant: \(13 - 2 = 11\)
Therefore, the resulting difference is: \[ 6x^2 - 16x + 11 \]
So the completed polynomial equation is: \[ 9x^2 + x + 13 - 3x^2 - 17x - 2 = 6x^2 - 16x + 11 \]
The blank would be filled with "11" for the constant term in the second expression.
Thus, the completed polynomial equation is: \[ 9x^2 + x + 13 - (2x^2 + 12x + 2) = 6x^2 - 16x + 11 \]
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