Question

Fill in the blanks to complete the polynomial equation that shows the subtraction of the second expression from the first and the resulting difference.
9x^2+_x +13
-2x^2+12x+_
_x^2+5x+2

Answer 1

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:

1. Let the first blank in the first polynomial be \( a \).
2. Let the second blank in the second polynomial be \( b \).
3. 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:

1. From the \( x^2 \) terms: \( 7 = c \)
2. From the \( x \) terms: \( a - 12 = 5 \)
3. From the constant terms: \( 13 - b = 2 \)

Now we solve each of these equations:

1. Since \( 7 = c \), we have \( c = 7 \).
2. For \( a - 12 = 5 \):
\[
a = 5 + 12 = 17
\]
3. 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**.