High School

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Simplify the following expression:

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

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

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

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

D. [tex]10x^4 + x^3 + x^2 + 4x[/tex]

Answer :

We start with the expression

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

Step 1. Group like terms

Arrange the terms by their exponents:

- For [tex]\(x^5\)[/tex]: [tex]\(7x^5\)[/tex].
- For [tex]\(x^4\)[/tex]: [tex]\(3x^4 - 3x^4\)[/tex].
- For [tex]\(x^3\)[/tex]: [tex]\(2x^3 - 3x^3\)[/tex].
- For [tex]\(x^2\)[/tex]: [tex]\(-5x^2 + 4x^2\)[/tex].
- For [tex]\(x\)[/tex]: [tex]\(6x - 2x\)[/tex].

Step 2. Combine the grouped terms

1. [tex]\(x^5\)[/tex] term:
[tex]$$
7x^5.
$$[/tex]

2. [tex]\(x^4\)[/tex] terms:
[tex]$$
3x^4 - 3x^4 = 0.
$$[/tex]

3. [tex]\(x^3\)[/tex] terms:
[tex]$$
2x^3 - 3x^3 = -x^3.
$$[/tex]

4. [tex]\(x^2\)[/tex] terms:
[tex]$$
-5x^2 + 4x^2 = -x^2.
$$[/tex]

5. [tex]\(x\)[/tex] terms:
[tex]$$
6x - 2x = 4x.
$$[/tex]

Step 3. Write the simplified expression

Adding all the combined terms together we obtain

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

Thus, the expression simplifies to

[tex]$$
\boxed{7x^5 - x^3 - x^2 + 4x}.
$$[/tex]

This corresponds to option A.

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