College

Simplify:

[tex]\left(x^5 - 2x^3 + x^2 - 7 \right) - \left( 2x^5 + 7x^4 - 4x^3 + 2 \right)[/tex]

A. [tex] -x^5 - 7x^4 + 2x^3 + x^2 - 5 [/tex]

B. [tex] -x^5 - 7x^4 - 6x^3 + x^2 - 5 [/tex]

C. [tex] -x^5 - 7x^4 + 2x^3 + x^2 - 9 [/tex]

D. [tex] -x^5 + 7x^4 - 6x^3 + x^2 - 5 [/tex]

Answer :

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]

C. [tex]-x^5 - 7x^4 + 2x^3 + x^2 - 9[/tex] is the simplified expression

1. Distribute the negative sign in the second set of parentheses:

[tex]x^5 - 2x^3 + x^2 - 7 - (2x^5 + 7x^4 - 4x^3 + 2).[/tex]

This becomes:

[tex]x^5 - 2x^3 + x^2 - 7 - 2x^5 - 7x^4 + 4x^3 - 2.[/tex]

2. Combine like terms:

Combine the [tex]x^5[/tex] terms:
[tex]x^5 - 2x^5 = -x^5.[/tex]

Combine the [tex]x^4[/tex] terms:
[tex]0 - 7x^4 = -7x^4.[/tex]

Combine the [tex]x^3[/tex] terms:
[tex]-2x^3 + 4x^3 = 2x^3.[/tex]

Combine the [tex]x^2[/tex] terms:
[tex]x^2.[/tex]

Combine the constant terms:
[tex]-7 - 2 = -9.[/tex]

So, putting it all together, we get:

[tex]-x^5 - 7x^4 + 2x^3 + x^2 - 9.[/tex]

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