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 find the missing terms in the polynomial equation, we start with the subtraction of the second expression from the first. We have the following two expressions:

1. \(9x^2 + \_\_x + 13\)
2. \(-2x^2 + 12x + \_\_\)

We want to subtract the second expression from the first:

\[
(9x^2 + \_\_x + 13) - (-2x^2 + 12x + \_\_) = \_\_x^2 + 5x + 2
\]

When we simplify the left side, we get:

\[
9x^2 + \_\_x + 13 + 2x^2 - 12x - \_\_
\]

This simplifies to:

\[
(9x^2 + 2x^2) + (\_\_x - 12x) + (13 - \_\_)
\]

This results in:

\[
(11x^2) + (\_\_ - 12)x + (13 - \_\_)
\]

Now, we want this to equal the expression on the right side:

\[
\_\_x^2 + 5x + 2
\]

From this, we can set up equations based on the coefficients:

1. For \(x^2\):
\[
11 = \_\_ \quad \Rightarrow \quad \_\_ = 11
\]

2. For \(x\):
\[
\_\_ - 12 = 5 \quad \Rightarrow \quad \_\_ = 5 + 12 = 17
\]

3. For the constant term:
\[
13 - \_\_ = 2 \quad \Rightarrow \quad \_\_ = 13 - 2 = 11
\]

Putting it all together, we have:

- The missing coefficient for \(x\) in the first expression is **17**.
- The missing term in the second expression is **11**.
- The missing coefficient for the constant term in the second expression is **11**.

So, the completed polynomial equation is:

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

Final answers:
- **17** for the \(x\) term in the first expression.
- **11** for the constant term in the second expression.
- **11** for the \(x^2\) term in the resulting difference.