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Answer :
To find a sixth-degree polynomial with a leading coefficient of seven and a constant term of four, let's analyze each option:
A. [tex]\(6x^7 - x^5 + 2x + 4\)[/tex]
- The highest degree term here is [tex]\(6x^7\)[/tex], which indicates this is a seventh-degree polynomial. We are looking for a sixth-degree polynomial, so this option doesn't meet the criteria.
B. [tex]\(4 + x + 7x^6 - 3x^2\)[/tex]
- The highest degree term here is [tex]\(7x^6\)[/tex], which means this is a sixth-degree polynomial. The leading coefficient is 7, and the constant term is 4. This matches all the given conditions.
C. [tex]\(7x^4 + 6 + x^2\)[/tex]
- The highest degree term here is [tex]\(7x^4\)[/tex], making this a fourth-degree polynomial. We need a sixth-degree polynomial, so this option is not correct.
D. [tex]\(5x + 4x^6 + 7\)[/tex]
- The highest degree term here is [tex]\(4x^6\)[/tex]. While this is a sixth-degree polynomial, its leading coefficient is 4, not 7. Therefore, this option does not satisfy all the conditions.
Based on this analysis, option B is the correct choice for a sixth-degree polynomial with a leading coefficient of seven and a constant term of four.
A. [tex]\(6x^7 - x^5 + 2x + 4\)[/tex]
- The highest degree term here is [tex]\(6x^7\)[/tex], which indicates this is a seventh-degree polynomial. We are looking for a sixth-degree polynomial, so this option doesn't meet the criteria.
B. [tex]\(4 + x + 7x^6 - 3x^2\)[/tex]
- The highest degree term here is [tex]\(7x^6\)[/tex], which means this is a sixth-degree polynomial. The leading coefficient is 7, and the constant term is 4. This matches all the given conditions.
C. [tex]\(7x^4 + 6 + x^2\)[/tex]
- The highest degree term here is [tex]\(7x^4\)[/tex], making this a fourth-degree polynomial. We need a sixth-degree polynomial, so this option is not correct.
D. [tex]\(5x + 4x^6 + 7\)[/tex]
- The highest degree term here is [tex]\(4x^6\)[/tex]. While this is a sixth-degree polynomial, its leading coefficient is 4, not 7. Therefore, this option does not satisfy all the conditions.
Based on this analysis, option B is the correct choice for a sixth-degree polynomial with a leading coefficient of seven and a constant term of four.
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