To complete the polynomial equation given, we need to find values for the blanks such that:
\[ (9x^2 + _x + 13) - (2x^2 + 12x + _) = _x^2 + 5x + 2 \]
Let's denote the missing terms with variables for clarity:
- Let the first blank in the first polynomial be \( a \).
- Let the second blank in the second polynomial be \( b \).
- Let the blank in the resulting polynomial be \( c \).
This gives us:
\[ (9x^2 + ax + 13) - (2x^2 + 12x + b) = cx^2 + 5x + 2 \]
Now, we distribute the negative sign through the second polynomial:
\[ 9x^2 + ax + 13 - 2x^2 - 12x - b = cx^2 + 5x + 2 \]
Combining like terms on the left side results in:
\[ (9x^2 - 2x^2) + (ax - 12x) + (13 - b) = cx^2 + 5x + 2 \]
This simplifies to:
\[ (7x^2 + (a - 12)x + (13 - b)) = cx^2 + 5x + 2 \]
From this equation, we can set up the following for the coefficients:
- From the \( x^2 \) terms: \( 7 = c \)
- From the \( x \) terms: \( a - 12 = 5 \)
- From the constant terms: \( 13 - b = 2 \)
Now we solve each of these equations:
- Since \( 7 = c \), we have \( c = 7 \).
- For \( a - 12 = 5 \): \[ a = 5 + 12 = 17 \]
- For \( 13 - b = 2 \): \[ b = 13 - 2 = 11 \]
Thus, we find:
- The first blank (coefficient of \( x \) in the first polynomial) is \( 17 \).
- The second blank (constant term of the second polynomial) is \( 11 \).
- The blank in the resulting polynomial (coefficient of \( x^2 \)) is \( 7 \).
Putting it all together, the complete equation becomes:
\[ 9x^2 + 17x + 13 - (2x^2 + 12x + 11) = 7x^2 + 5x + 2 \]
In summary:
- The first blank is 17.
- The second blank is 11.
- The third blank is 7.