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------------------------------------------------ Select all polynomial expressions that are equivalent to [tex]6x^4 + 4x^3 - 7x^2 + 5x + 8[/tex].

A. [tex]16x^{10}[/tex]

B. [tex]6x^5 + 4x^4 - 7x^3 + 5x^2 + 8x[/tex]

C. [tex]6x^4 + 4x^3 - 7x^2 + 5x + 8[/tex]

D. [tex]8 + 5x + 7x^2 - 4x^3 + 6x^4[/tex]

E. [tex]8 + 5x - 7x^2 + 4x^3 + 6x^4[/tex]

Answer :

To determine which polynomial expressions are equivalent to the given polynomial [tex]\(6x^4 + 4x^3 - 7x^2 + 5x + 8\)[/tex], we need to compare each provided option with the given polynomial.

Let's analyze each option:

1. Option A: [tex]\(16x^{10}\)[/tex]

This expression has an entirely different degree and format compared to the given polynomial. It is not equivalent.

2. Option B: [tex]\(6x^5 + 4x^4 - 7x^3 + 5x^2 + 8x\)[/tex]

This expression has different exponents for each term compared to the given polynomial. It is not equivalent.

3. Option C: [tex]\(6x^4 + 4x^3 - 7x^2 + 5x + 8\)[/tex]

This expression is exactly the same as the given polynomial. Therefore, it is equivalent.

4. Option D: [tex]\(8 + 5x + 7x^2 - 4x^3 + 6x^4\)[/tex]

Although this expression contains all the same terms, they are in a different order. Remember that polynomials are often reordered in standard form with terms from highest to lowest degree. Reordering this expression gives:

[tex]\( 6x^4 - 4x^3 + 7x^2 + 5x + 8 \)[/tex]

This does not match the given polynomial because signs of some terms are different. So, it is not equivalent.

5. Option E: [tex]\(8 + 5x - 7x^2 + 4x^3 + 6x^4\)[/tex]

Similarly, this expression also contains all the same terms but reordered. Let's reorder:

[tex]\(6x^4 + 4x^3 - 7x^2 + 5x + 8\)[/tex]

This matches exactly with the given polynomial. Therefore, this expression is equivalent.

Therefore, based on the comparisons, the polynomial expressions that are equivalent to [tex]\(6x^4 + 4x^3 - 7x^2 + 5x + 8\)[/tex] are:

- Option C: [tex]\(6x^4 + 4x^3 - 7x^2 + 5x + 8\)[/tex]
- Option E: [tex]\(8 + 5x - 7x^2 + 4x^3 + 6x^4\)[/tex]

Thus, the correct answers are:
- Option C
- Option E