To simplify the expression [tex]\(\left( x^5 - 2x^3 + x^2 - 7\right) - \left(2x^5 + 7x^4 - 4x^3 + 2\right)\)[/tex], follow these steps:
1. Distribute the negative sign through the second polynomial:
[tex]\[
\left( x^5 - 2x^3 + x^2 - 7 \right) - \left( 2x^5 + 7x^4 - 4x^3 + 2 \right)
\][/tex]
[tex]\[
= x^5 - 2x^3 + x^2 - 7 - 2x^5 - 7x^4 + 4x^3 - 2
\][/tex]
2. Combine like terms:
- For [tex]\(x^5\)[/tex]:
[tex]\[
x^5 - 2x^5 = -x^5
\][/tex]
- For [tex]\(x^4\)[/tex]:
[tex]\[
-7x^4
\][/tex]
- For [tex]\(x^3\)[/tex]:
[tex]\[
-2x^3 + 4x^3 = 2x^3
\][/tex]
- For [tex]\(x^2\)[/tex]:
[tex]\[
x^2
\][/tex]
- Constant terms:
[tex]\[
-7 - 2 = -9
\][/tex]
3. Combine all the simplified terms:
[tex]\[
-x^5 - 7x^4 + 2x^3 + x^2 - 9
\][/tex]
Therefore, the simplified expression is:
[tex]\[
-x^5 - 7x^4 + 2x^3 + x^2 - 9
\][/tex]
So, the correct answer is:
[tex]\[
-x^5 - 7x^4 + 2x^3 + x^2 - 9
\][/tex]
