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, 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:

1. \( d = 11 \)
2. \( b - 12 = 5 \)
3. \( 13 - c = 2 \)

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:

\( 9x^2 + 17x + 13 \)
\( -2x^2 + 12x + 11 \)
\( 11x^2 + 5x + 2 \)