Answer :
We are asked to find a sixth-degree polynomial with a leading coefficient of 7 and a constant term of 4 among the following candidates:
1. [tex]$$6x^7 - x^5 + 2x + 4$$[/tex]
2. [tex]$$4 + x + 7x^6 - 3x^2$$[/tex]
3. [tex]$$7x^4 + 6 + x^2$$[/tex]
4. [tex]$$5x + 4x^6 + 7$$[/tex]
Let’s analyze each candidate step by step.
1. For the polynomial
[tex]$$6x^7 - x^5 + 2x + 4,$$[/tex]
the term with the highest exponent is [tex]$$6x^7.$$[/tex] This means the degree is 7 with a leading coefficient of 6. Also, the constant term is 4. However, since the degree is 7 instead of 6, this candidate does not meet the requirement.
2. For the polynomial
[tex]$$4 + x + 7x^6 - 3x^2,$$[/tex]
if we rearrange it in standard form, we get
[tex]$$7x^6 - 3x^2 + x + 4.$$[/tex]
Here, the highest-degree term is [tex]$$7x^6,$$[/tex] so the degree is 6 and the leading coefficient is 7. The constant term is 4. All conditions are met.
3. For the polynomial
[tex]$$7x^4 + 6 + x^2,$$[/tex]
rearranged as
[tex]$$7x^4 + x^2 + 6,$$[/tex]
the highest-degree term is [tex]$$7x^4.$$[/tex] This means the degree is 4, which is not the required 6. Thus, this candidate does not qualify.
4. For the polynomial
[tex]$$5x + 4x^6 + 7,$$[/tex]
rearranged as
[tex]$$4x^6 + 5x + 7,$$[/tex]
the highest-degree term is [tex]$$4x^6.$$[/tex] So the degree is 6 but the leading coefficient is 4 (not 7) and the constant term is 7 (not 4). This candidate also does not meet the conditions.
Because only the second polynomial has a degree of 6, a leading coefficient of 7, and a constant term of 4, the correct answer is candidate 2.
Thus, the correct answer is:
[tex]$$\boxed{2}$$[/tex]
1. [tex]$$6x^7 - x^5 + 2x + 4$$[/tex]
2. [tex]$$4 + x + 7x^6 - 3x^2$$[/tex]
3. [tex]$$7x^4 + 6 + x^2$$[/tex]
4. [tex]$$5x + 4x^6 + 7$$[/tex]
Let’s analyze each candidate step by step.
1. For the polynomial
[tex]$$6x^7 - x^5 + 2x + 4,$$[/tex]
the term with the highest exponent is [tex]$$6x^7.$$[/tex] This means the degree is 7 with a leading coefficient of 6. Also, the constant term is 4. However, since the degree is 7 instead of 6, this candidate does not meet the requirement.
2. For the polynomial
[tex]$$4 + x + 7x^6 - 3x^2,$$[/tex]
if we rearrange it in standard form, we get
[tex]$$7x^6 - 3x^2 + x + 4.$$[/tex]
Here, the highest-degree term is [tex]$$7x^6,$$[/tex] so the degree is 6 and the leading coefficient is 7. The constant term is 4. All conditions are met.
3. For the polynomial
[tex]$$7x^4 + 6 + x^2,$$[/tex]
rearranged as
[tex]$$7x^4 + x^2 + 6,$$[/tex]
the highest-degree term is [tex]$$7x^4.$$[/tex] This means the degree is 4, which is not the required 6. Thus, this candidate does not qualify.
4. For the polynomial
[tex]$$5x + 4x^6 + 7,$$[/tex]
rearranged as
[tex]$$4x^6 + 5x + 7,$$[/tex]
the highest-degree term is [tex]$$4x^6.$$[/tex] So the degree is 6 but the leading coefficient is 4 (not 7) and the constant term is 7 (not 4). This candidate also does not meet the conditions.
Because only the second polynomial has a degree of 6, a leading coefficient of 7, and a constant term of 4, the correct answer is candidate 2.
Thus, the correct answer is:
[tex]$$\boxed{2}$$[/tex]
