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Answer :
To find an example of a sixth-degree polynomial with a leading coefficient of seven and a constant term of four, we need to analyze each of the given options. Let's break it down:
1. Option (1): [tex]\(6x^7 - x^5 + 2x + 4\)[/tex]
- The highest degree term here is [tex]\(6x^7\)[/tex], which is a seventh-degree term. We're looking for a sixth-degree polynomial, so this option is not correct.
2. Option (3): [tex]\(7x^4 + 6 + x^2\)[/tex]
- The highest degree term here is [tex]\(7x^4\)[/tex], which is a fourth-degree term. This means it does not have a sixth-degree term at all, so this option is not correct.
3. Option (2): [tex]\(4 + x + 7x^6 - 3x^2\)[/tex]
- The highest degree term here is [tex]\(7x^6\)[/tex], which makes it a sixth-degree polynomial. The leading coefficient of this sixth-degree term is 7, which matches our requirement.
- The constant term here is 4, which also matches our requirement.
4. Option (4): [tex]\(5x + 4x^6 + 7\)[/tex]
- The highest degree term here is [tex]\(4x^6\)[/tex], which is a sixth-degree term. However, the leading coefficient of this term is 4, and we need the leading coefficient to be 7. Therefore, this option is not correct.
Based on the analysis, Option (2) is the correct choice because it is a sixth-degree polynomial with a leading coefficient of seven and a constant term of four.
1. Option (1): [tex]\(6x^7 - x^5 + 2x + 4\)[/tex]
- The highest degree term here is [tex]\(6x^7\)[/tex], which is a seventh-degree term. We're looking for a sixth-degree polynomial, so this option is not correct.
2. Option (3): [tex]\(7x^4 + 6 + x^2\)[/tex]
- The highest degree term here is [tex]\(7x^4\)[/tex], which is a fourth-degree term. This means it does not have a sixth-degree term at all, so this option is not correct.
3. Option (2): [tex]\(4 + x + 7x^6 - 3x^2\)[/tex]
- The highest degree term here is [tex]\(7x^6\)[/tex], which makes it a sixth-degree polynomial. The leading coefficient of this sixth-degree term is 7, which matches our requirement.
- The constant term here is 4, which also matches our requirement.
4. Option (4): [tex]\(5x + 4x^6 + 7\)[/tex]
- The highest degree term here is [tex]\(4x^6\)[/tex], which is a sixth-degree term. However, the leading coefficient of this term is 4, and we need the leading coefficient to be 7. Therefore, this option is not correct.
Based on the analysis, Option (2) is the correct choice because it is a sixth-degree polynomial with a leading coefficient of seven and a constant term of four.
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