Question

Which of the following correctly determines the first and last terms of the expression (12x5+4x4+9x3−10x2+15)−(24x5+9x3−7x2+8)?(1 point)
Responses

The first term is −12x5, and the last term is 7.
The first term is negative 12 x superscript 5 baseline , and the last term is 7.

The first term is 12x5, and the last term is −7.
The first term is 12 x superscript 5 baseline , and the last term is negative 7 .

The first term is 4x4 , and the last term is −3x2.
The first term is 4 x superscript 4 baseline , and the last term is negative 3 x squared .

The first term is −12, and the last term is 7.

Answer 1

To determine the first and last terms of the expression \((12x^5 + 4x^4 + 9x^3 - 10x^2 + 15) - (24x^5 + 9x^3 - 7x^2 + 8)\), we first need to simplify the expression by distributing the negative sign and combining like terms.

Starting with the expression:

\[
(12x^5 + 4x^4 + 9x^3 - 10x^2 + 15) - (24x^5 + 9x^3 - 7x^2 + 8)
\]

Distributing the negative:

\[
12x^5 + 4x^4 + 9x^3 - 10x^2 + 15 - 24x^5 - 9x^3 + 7x^2 - 8
\]

Now, combine the like terms:

1. **For \(x^5\) terms**: \(12x^5 - 24x^5 = -12x^5\)
2. **For \(x^4\) terms**: \(4x^4\)
3. **For \(x^3\) terms**: \(9x^3 - 9x^3 = 0\)
4. **For \(x^2\) terms**: \(-10x^2 + 7x^2 = -3x^2\)
5. **For constant terms**: \(15 - 8 = 7\)

Putting it all together, we get:

\[
-12x^5 + 4x^4 - 3x^2 + 7
\]

Now, we identify the first and last terms of the resulting polynomial:

- The **first term** (highest degree) is \(-12x^5\).
- The **last term** (constant term) is \(7\).

Therefore, the correct choice is:

**The first term is \(-12x^5\), and the last term is \(7\).**