To complete the polynomial equation, we need to subtract the second expression from the first and ensure that the resulting difference matches the third expression.
Let's set up the expression:
1st polynomial: \( 9x^2 + bx + 13 \)
2nd polynomial: \( -2x^2 + 12x + c \)
3rd polynomial (result): \( dx^2 + 5x + 2 \)
Accordingly, we will subtract the second polynomial from the first:
\[ (9x^2 + bx + 13) - (-2x^2 + 12x + c) = (9x^2 + bx + 13) + (2x^2 - 12x - c) \]
Now, let's combine like terms:
\[ (9x^2 + 2x^2) + (bx - 12x) + (13 - c) = (9 + 2)x^2 + (b - 12)x + (13 - c) \]
This leads to:
\[ 11x^2 + (b - 12)x + (13 - c) \]
We want the resulting polynomial to match:
\[ dx^2 + 5x + 2 \]
Comparing coefficients, we get the following equations:
- \( d = 11 \)
- \( b - 12 = 5 \)
- \( 13 - c = 2 \)
From \( d = 11 \), we have \( d = 11 \).
From \( b - 12 = 5 \): \[ b = 5 + 12 = 17 \]
From \( 13 - c = 2 \): \[ c = 13 - 2 = 11 \]
So, we can fill in the blanks with:
- For \( bx \): \( 17x \)
- For \( c \): \( 11 \)
Thus, the completed polynomial equation is:
\[ 9x^2 + 17x + 13 \] \[
- ( -2x^2 + 12x + 11) \] \[ = 11x^2 + 5x + 2 \]
So the filled polynomial is:
\( 9x^2 + 17x + 13 \)
\( -2x^2 + 12x + 11 \)
\( 11x^2 + 5x + 2 \)
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