Question

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

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 −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 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.

1. Distribute the negative sign to the second polynomial:

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

2. Combine like terms:

- For \(x^5\): \(12x^5 - 24x^5 = -12x^5\)
- For \(x^4\): \(4x^4\) (only one term)
- For \(x^3\): \(9x^3 - 9x^3 = 0\)
- For \(x^2\): \(-10x^2 + 7x^2 = -3x^2\)
- For the constant term: \(15 - 8 = 7\)

So the combined expression is:

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

3. Identify the first and last terms from the simplified expression:

- The **first term** (the term with the highest power of \(x\)) is \(-12x^5\).
- The **last term** (the constant term) is \(7\).

Therefore, the correct response is:

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