High School

Simplify the following expression:

\[ 3x^4 + 2x^3 - 5x^2 + 4x^2 + 6x - 2x - 3x^4 + 7x^5 - 3x^3 \]

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

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

C. \[ 10x^4 + x^3 + x^2 + 4x \]

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

Answer :

To simplify the expression [tex]\(3x^4 + 2x^3 - 5x^2 + 4x^2 + 6x - 2x - 3x^4 + 7x^5 - 3x^3\)[/tex], follow these steps:

1. Combine Like Terms:
- Start by grouping the terms that have the same power of [tex]\(x\)[/tex].

2. Identifying Terms:
- [tex]\(7x^5\)[/tex] is the only term with [tex]\(x^5\)[/tex].
- For [tex]\(x^4\)[/tex]: [tex]\(3x^4\)[/tex] and [tex]\(-3x^4\)[/tex].
- For [tex]\(x^3\)[/tex]: [tex]\(2x^3\)[/tex] and [tex]\(-3x^3\)[/tex].
- For [tex]\(x^2\)[/tex]: [tex]\(-5x^2\)[/tex] and [tex]\(4x^2\)[/tex].
- For [tex]\(x\)[/tex]: [tex]\(6x\)[/tex] and [tex]\(-2x\)[/tex].

3. Simplify Each Group:
- [tex]\(x^5\)[/tex] terms: [tex]\(7x^5\)[/tex].
- [tex]\(x^4\)[/tex] terms: [tex]\(3x^4 - 3x^4 = 0\)[/tex].
- [tex]\(x^3\)[/tex] terms: [tex]\(2x^3 - 3x^3 = -x^3\)[/tex].
- [tex]\(x^2\)[/tex] terms: [tex]\(-5x^2 + 4x^2 = -x^2\)[/tex].
- [tex]\(x\)[/tex] terms: [tex]\(6x - 2x = 4x\)[/tex].

4. Write the Simplified Expression:
- Combine the results from each group: [tex]\(7x^5 - x^3 - x^2 + 4x\)[/tex].

The final simplified expression is:
[tex]\[ 7x^5 - x^3 - x^2 + 4x \][/tex]

Therefore, the correct answer is:
A. [tex]\(7x^5 - x^3 - x^2 + 4x\)[/tex]

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