Answer :
To find a sixth-degree polynomial with a leading coefficient of seven and a constant term of four, let's evaluate each option step by step:
1. Option (1): [tex]\(6x^7 - x^5 + 2x + 4\)[/tex]
- This polynomial has a degree of 7 (highest exponent is 7), which does not match our requirement of a sixth-degree polynomial. Therefore, this option is incorrect.
2. Option (2): [tex]\(4 + x + 7x^6 - 3x^2\)[/tex]
- This polynomial is of degree 6 because the highest exponent is 6.
- The leading term is [tex]\(7x^6\)[/tex], which means the leading coefficient is 7, as desired.
- The constant term here is 4, matching our requirement.
- This option satisfies all the conditions for the polynomial we are looking for.
3. Option (3): [tex]\(7x^4 + 6 + x^2\)[/tex]
- This polynomial is of degree 4 (highest exponent is 4), not the required degree of 6. It is not a sixth-degree polynomial, so this option is incorrect.
4. Option (4): [tex]\(5x + 4x^6 + 7\)[/tex]
- This polynomial is of degree 6 (highest exponent is 6).
- However, the leading term is [tex]\(4x^6\)[/tex], meaning the leading coefficient is 4, not 7. Therefore, this option does not meet the given criteria.
Based on these evaluations, the correct choice is Option (2): [tex]\(4 + x + 7x^6 - 3x^2\)[/tex]. This polynomial is of degree 6, with a leading coefficient of 7 and a constant term of 4.
1. Option (1): [tex]\(6x^7 - x^5 + 2x + 4\)[/tex]
- This polynomial has a degree of 7 (highest exponent is 7), which does not match our requirement of a sixth-degree polynomial. Therefore, this option is incorrect.
2. Option (2): [tex]\(4 + x + 7x^6 - 3x^2\)[/tex]
- This polynomial is of degree 6 because the highest exponent is 6.
- The leading term is [tex]\(7x^6\)[/tex], which means the leading coefficient is 7, as desired.
- The constant term here is 4, matching our requirement.
- This option satisfies all the conditions for the polynomial we are looking for.
3. Option (3): [tex]\(7x^4 + 6 + x^2\)[/tex]
- This polynomial is of degree 4 (highest exponent is 4), not the required degree of 6. It is not a sixth-degree polynomial, so this option is incorrect.
4. Option (4): [tex]\(5x + 4x^6 + 7\)[/tex]
- This polynomial is of degree 6 (highest exponent is 6).
- However, the leading term is [tex]\(4x^6\)[/tex], meaning the leading coefficient is 4, not 7. Therefore, this option does not meet the given criteria.
Based on these evaluations, the correct choice is Option (2): [tex]\(4 + x + 7x^6 - 3x^2\)[/tex]. This polynomial is of degree 6, with a leading coefficient of 7 and a constant term of 4.
