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 + _
_x2 + 5x + 2

Answer 1

To fill in the blanks and complete the polynomial equation, we start by performing the subtraction of the second expression from the first:

First expression: \( 9x^2 + bx + 13 \)
Second expression: \( 2x^2 + 12x + c \)

We want to find the resulting polynomial difference:
\[
(9x^2 + bx + 13) - (2x^2 + 12x + c) = (9x^2 - 2x^2) + (bx - 12x) + (13 - c)
\]
This simplifies to:
\[
(7x^2) + (b - 12)x + (13 - c)
\]

We know from the resulting expression given in the problem:
\[
x^2 + 5x + 2
\]
Now we can set up equations by comparing coefficients:

1. Coefficient of \(x^2\):
\[
7 = 1 \quad \text{(which is not true, so we have a misunderstanding here about what is compared)}
\]
It appears we stated this wrong. We are actually looking at this from what the resulting expression is.

Here we need:
1. We need the coefficient of \(x^2\) from the resulting difference to equal 1.
Hence:
\[
7 = 1 \implies \text{This is wrong! Let's solve for others first!}
\]

2. Coefficient of \(x\):
\[
b - 12 = 5 \implies b = 17
\]

3. Constant term:
\[
13 - c = 2 \implies c = 11
\]

Now we can fill in the blanks:

The completed polynomial will look like this:
\[
9x^2 + 17x + 13 - (2x^2 + 12x + 11) = 7x^2 + 5x + 2
\]
Thus, the blanks filled in yield:
- \( b = 17 \)
- \( c = 11 \)

So the completed polynomial equation with blanks filled in is:
\[
9x^2 + 17x + 13 - 2x^2 + 12x + 11 = 7x^2 + 5x + 2
\]