To complete the polynomial equation showing the subtraction of the second expression from the first, we need to determine how to subtract the second polynomial from the first. Here are the steps outlined:
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Identify the polynomials:
- First polynomial: \( 9x^2 + bx + 13 \)
- Second polynomial: \( -(2x^2 + 12x + c) \)
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Set up the subtraction: \[ (9x^2 + bx + 13) - (2x^2 + 12x + c) = (9x^2 - 2x^2) + (bx - 12x) + (13 - c) \]
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Combine like terms:
- For the \( x^2 \) terms: \( 9x^2 - 2x^2 = 7x^2 \)
- For the \( x \) terms: \( (b - 12)x \)
- For the constant terms: \( 13 - c \)
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Given the resulting polynomial: The resulting polynomial is given as \( dx^2 + 5x + 2 \). We can compare coefficients:
- From \( 7x^2 = dx^2 \) --> \( d = 7 \)
- From \( (b - 12)x = 5x \) --> \( b - 12 = 5 \) --> \( b = 17 \)
- From the constant terms: \( 13 - c = 2 \) --> \( -c = 2 - 13 \) --> \( -c = -11 \) --> \( c = 11 \)
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Fill in the blanks:
- The value of \( b \) is \( 17 \).
- The value of \( c \) is \( 11 \).
So, the completed polynomial equation is:
\[ 9x^2 + 17x + 13 - (2x^2 + 12x + 11) = 7x^2 + 5x + 2 \]
Final answer: The blanks filled in are:
- 17
- 11
- 7
In summary: \[ 9x^2 + 17x + 13 - (2x^2 + 12x + 11) = 7x^2 + 5x + 2 \]
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