To complete the polynomial equation, we need to interpret the expression correctly. The goal is to subtract the second polynomial from the first polynomial.
Given:
- First polynomial: \( 9x^2 + ( )x + 13 \)
- Second polynomial: \( 2x^2 + 12x + ( )x^2 + 5x + 2 \)
Let's first combine similar terms of the second polynomial. We'll assume the second parentheses is also a placeholder for a coefficient of \( x^2 \):
Step 1: Combine the second polynomial's terms.
If we denote the unknown coefficient in the second polynomial as \( c \), the full second polynomial can be expressed as: \[ 2x^2 + cx^2 + 12x + 5x + 2 = (2 + c)x^2 + 17x + 2 \]
Step 2: Set Up the Polynomial Equation for Subtraction
Now, we are to subtract the second polynomial from the first: \[ 9x^2 + bx + 13 - [(2 + c)x^2 + 17x + 2] \]
Where \( b \) is the unknown coefficient for \( x \) in the first polynomial.
Step 3: Perform the Subtraction
Distributing the negative sign, we get: \[ 9x^2 + bx + 13 - (2 + c)x^2 - 17x - 2 \]
Now, combine like terms:
- For \( x^2 \): \[ (9 - (2 + c))x^2 = (9 - 2 - c)x^2 = (7 - c)x^2 \]
- For \( x \): \[ (b - 17)x \]
- For the constant: \[ 13 - 2 = 11 \]
Step 4: Form the Resulting Expression
Putting it all together, we have: \[ (7 - c)x^2 + (b - 17)x + 11 \]
Since we need to fill in the blanks in the original polynomial expression, we can summarize:
- For the \( x^2 \) term in the first polynomial: It would mean \( 7-c = 0 \) or simply \(c\) needs to be chosen accordingly as it doesn't have a specific value given.
- For the \( x \) term, we leave \(b\) without a predetermined value as it operates independently.
Given that, we can summarize that the blanks that need to be filled, depending on the choices for \( b \) and \( c \), would look something like this:
So the completed expression for subtraction can be stylized as follows:
9x^2 + (b)x + 13 - (2+c)x^2 - 17x - 2 => (7 - c)x^2 + (b - 17)x + 11
Thus, typically, common choices would be:
- \( (b = 17) \) and \( (c = 0) \) resulting in a dealt equation give simplifying to:
That means each blank could contain any integer relationship based on how the \( b \) and \( c \) values are aligned.
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