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Answer :
To solve the problem of determining which polynomial expressions are equivalent to [tex]\(6x^4 + 4x^3 - 7x^2 + 5x + 8\)[/tex], let's look at each given option and compare them to our reference polynomial.
1. The reference polynomial is:
[tex]\[
6x^4 + 4x^3 - 7x^2 + 5x + 8
\][/tex]
2. Let's check each of the given options one by one:
a. [tex]\(16x^{10}\)[/tex]
- This polynomial has a completely different degree (power) and does not match [tex]\(6x^4 + 4x^3 - 7x^2 + 5x + 8\)[/tex] at all.
- Not equivalent.
b. [tex]\(6x^5 + 4x^4 - 7x^3 + 5x^2 + 8x\)[/tex]
- The highest power here is [tex]\(x^5\)[/tex], and none of the powers of [tex]\(x\)[/tex] match the reference polynomial exactly.
- Not equivalent.
c. [tex]\(6x^4 + 8 + 4x^3 + 5x - 7x^2\)[/tex]
- Rearranging this polynomial:
[tex]\[
6x^4 + 4x^3 - 7x^2 + 5x + 8
\][/tex]
- This matches the reference polynomial exactly.
- Equivalent.
d. [tex]\(8 + 5x + 7x^2 - 4x^3 + 6x^4\)[/tex]
- Rearranging this polynomial:
[tex]\[
6x^4 - 4x^3 + 7x^2 + 5x + 8
\][/tex]
- While the coefficients are mostly the same, the signs and positions of some terms do not match.
- Not equivalent.
e. [tex]\(8 + 5x - 7x^2 + 4x^3 + 6x^4\)[/tex]
- Rearranging this polynomial:
[tex]\[
6x^4 + 4x^3 - 7x^2 + 5x + 8
\][/tex]
- This matches the reference polynomial exactly.
- Equivalent.
3. Based on our analysis, the polynomial expressions that are equivalent to [tex]\(6x^4 + 4x^3 - 7x^2 + 5x + 8\)[/tex] are:
- Option c
- Option e
Thus, the correct answers are:
[tex]\[
\boxed{c \text{ and } e}
\][/tex]
1. The reference polynomial is:
[tex]\[
6x^4 + 4x^3 - 7x^2 + 5x + 8
\][/tex]
2. Let's check each of the given options one by one:
a. [tex]\(16x^{10}\)[/tex]
- This polynomial has a completely different degree (power) and does not match [tex]\(6x^4 + 4x^3 - 7x^2 + 5x + 8\)[/tex] at all.
- Not equivalent.
b. [tex]\(6x^5 + 4x^4 - 7x^3 + 5x^2 + 8x\)[/tex]
- The highest power here is [tex]\(x^5\)[/tex], and none of the powers of [tex]\(x\)[/tex] match the reference polynomial exactly.
- Not equivalent.
c. [tex]\(6x^4 + 8 + 4x^3 + 5x - 7x^2\)[/tex]
- Rearranging this polynomial:
[tex]\[
6x^4 + 4x^3 - 7x^2 + 5x + 8
\][/tex]
- This matches the reference polynomial exactly.
- Equivalent.
d. [tex]\(8 + 5x + 7x^2 - 4x^3 + 6x^4\)[/tex]
- Rearranging this polynomial:
[tex]\[
6x^4 - 4x^3 + 7x^2 + 5x + 8
\][/tex]
- While the coefficients are mostly the same, the signs and positions of some terms do not match.
- Not equivalent.
e. [tex]\(8 + 5x - 7x^2 + 4x^3 + 6x^4\)[/tex]
- Rearranging this polynomial:
[tex]\[
6x^4 + 4x^3 - 7x^2 + 5x + 8
\][/tex]
- This matches the reference polynomial exactly.
- Equivalent.
3. Based on our analysis, the polynomial expressions that are equivalent to [tex]\(6x^4 + 4x^3 - 7x^2 + 5x + 8\)[/tex] are:
- Option c
- Option e
Thus, the correct answers are:
[tex]\[
\boxed{c \text{ and } e}
\][/tex]
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