Question

Fill in the blank to complete the polynomial equation that shows the subtraction of the second expression from the first and the resulting difference. (2 points)
(9x^2+__x+13)-(2x^2+12x+11)=(7x^2+5x+2)

Answer 1

To find the blank in the polynomial equation

\[
(9x^2 + \_ x + 13) - (2x^2 + 12x + 11) = (7x^2 + 5x + 2),
\]

we start by rewriting the left-hand side of the equation as:

\[
9x^2 + \_ x + 13 - 2x^2 - 12x - 11.
\]

Now, we will group like terms:

1. Combine the \(x^2\) terms:
\[
9x^2 - 2x^2 = 7x^2.
\]

2. Combine the \(x\) terms. We have \(\_ x - 12x\), which simplifies to \((\_ - 12)x\).

3. Combine the constant terms:
\[
13 - 11 = 2.
\]

Now, we can set up the equation from the left-hand side:

\[
(7x^2 + (\_ - 12)x + 2).
\]

This must equal the right-hand side:

\[
7x^2 + 5x + 2.
\]

From the \(x^2\) terms, we see they match. From the constant terms, \(2\) also matches. Now we can set the coefficients of \(x\) equal to each other:

\[
\_ - 12 = 5.
\]

To find \(\_\), we solve for it:

\[
\_ = 5 + 12 = 17.
\]

Therefore, the blank should be filled with \(17\). The complete equation is:

\[
(9x^2 + 17x + 13) - (2x^2 + 12x + 11) = (7x^2 + 5x + 2).
\]