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Answer :
To solve this problem, we're looking for a polynomial that satisfies a specific set of conditions: it must be a sixth-degree polynomial with a leading coefficient (highest degree term) of seven and a constant term of four. Let's examine each polynomial option provided:
1. Option 1: [tex]\(6x^7 - x^5 + 2x + 4\)[/tex]
- The highest degree term here is [tex]\(6x^7\)[/tex], which makes this a seventh-degree polynomial. Therefore, this option does not meet the requirement of being a sixth-degree polynomial.
2. Option 2: [tex]\(4 + x + 7x^6 - 3x^2\)[/tex]
- If we arrange the terms according to the degree (from highest to lowest), the polynomial is: [tex]\(7x^6 - 3x^2 + x + 4\)[/tex].
- The highest degree term is [tex]\(7x^6\)[/tex], which indicates this is a sixth-degree polynomial.
- The leading coefficient is 7, which matches the required condition.
- The constant term is 4, matching the required condition as well.
This option fits all the criteria.
3. Option 3: [tex]\(7x^4 + 6 + x^2\)[/tex]
- The highest degree term is [tex]\(7x^4\)[/tex], making this a fourth-degree polynomial, not a sixth-degree one. Therefore, this option does not meet the requirement.
4. Option 4: [tex]\(5x + 4x^6 + 7\)[/tex]
- Rearranging the terms, the polynomial is: [tex]\(4x^6 + 5x + 7\)[/tex].
- The highest degree term is [tex]\(4x^6\)[/tex], which makes it a sixth-degree polynomial.
- However, the leading coefficient is 4, not 7, so this option does not fulfill the leading coefficient requirement.
Based on these analyses, Option 2 is the polynomial that satisfies all the given criteria: it is a sixth-degree polynomial with a leading coefficient of 7 and a constant term of 4.
1. Option 1: [tex]\(6x^7 - x^5 + 2x + 4\)[/tex]
- The highest degree term here is [tex]\(6x^7\)[/tex], which makes this a seventh-degree polynomial. Therefore, this option does not meet the requirement of being a sixth-degree polynomial.
2. Option 2: [tex]\(4 + x + 7x^6 - 3x^2\)[/tex]
- If we arrange the terms according to the degree (from highest to lowest), the polynomial is: [tex]\(7x^6 - 3x^2 + x + 4\)[/tex].
- The highest degree term is [tex]\(7x^6\)[/tex], which indicates this is a sixth-degree polynomial.
- The leading coefficient is 7, which matches the required condition.
- The constant term is 4, matching the required condition as well.
This option fits all the criteria.
3. Option 3: [tex]\(7x^4 + 6 + x^2\)[/tex]
- The highest degree term is [tex]\(7x^4\)[/tex], making this a fourth-degree polynomial, not a sixth-degree one. Therefore, this option does not meet the requirement.
4. Option 4: [tex]\(5x + 4x^6 + 7\)[/tex]
- Rearranging the terms, the polynomial is: [tex]\(4x^6 + 5x + 7\)[/tex].
- The highest degree term is [tex]\(4x^6\)[/tex], which makes it a sixth-degree polynomial.
- However, the leading coefficient is 4, not 7, so this option does not fulfill the leading coefficient requirement.
Based on these analyses, Option 2 is the polynomial that satisfies all the given criteria: it is a sixth-degree polynomial with a leading coefficient of 7 and a constant term of 4.
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