Answer :
To identify the sixth-degree polynomial with a leading coefficient of seven and a constant term of four, we need to examine the given options based on these two criteria:
1. The polynomial must have a degree of 6. This means the highest power of [tex]\( x \)[/tex] should be [tex]\( x^6 \)[/tex].
2. The leading coefficient, which is the coefficient of the term with the highest power, must be 7.
3. The constant term, which is the term without [tex]\( x \)[/tex], should be 4.
Let's examine each option:
1) [tex]\(6x^7 - x^5 + 2x + 4\)[/tex]
- The highest degree term is [tex]\(6x^7\)[/tex], which means this is a seventh-degree polynomial, not sixth-degree.
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 (the coefficient of [tex]\(x^6\)[/tex]) is 7.
- The constant term is 4.
- This option satisfies all the requirements.
3) [tex]\(7x^4 + 6 + x^7\)[/tex]
- The highest degree term is [tex]\(x^7\)[/tex], making it a seventh-degree polynomial, not sixth-degree.
4) [tex]\(5x + 4x^6 + 7\)[/tex]
- Here, the highest degree term is [tex]\(4x^6\)[/tex], so it is a sixth-degree polynomial.
- The leading coefficient is 4, which is not what we're looking for because we need a leading coefficient of 7.
- The constant term is 7, not 4.
Based on the examinations above, option 2, [tex]\(4 + x + 7x^6 - 3x^2\)[/tex], is the correct example of a sixth-degree polynomial with a leading coefficient of seven and a constant term of four.
1. The polynomial must have a degree of 6. This means the highest power of [tex]\( x \)[/tex] should be [tex]\( x^6 \)[/tex].
2. The leading coefficient, which is the coefficient of the term with the highest power, must be 7.
3. The constant term, which is the term without [tex]\( x \)[/tex], should be 4.
Let's examine each option:
1) [tex]\(6x^7 - x^5 + 2x + 4\)[/tex]
- The highest degree term is [tex]\(6x^7\)[/tex], which means this is a seventh-degree polynomial, not sixth-degree.
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 (the coefficient of [tex]\(x^6\)[/tex]) is 7.
- The constant term is 4.
- This option satisfies all the requirements.
3) [tex]\(7x^4 + 6 + x^7\)[/tex]
- The highest degree term is [tex]\(x^7\)[/tex], making it a seventh-degree polynomial, not sixth-degree.
4) [tex]\(5x + 4x^6 + 7\)[/tex]
- Here, the highest degree term is [tex]\(4x^6\)[/tex], so it is a sixth-degree polynomial.
- The leading coefficient is 4, which is not what we're looking for because we need a leading coefficient of 7.
- The constant term is 7, not 4.
Based on the examinations above, option 2, [tex]\(4 + x + 7x^6 - 3x^2\)[/tex], is the correct example of a sixth-degree polynomial with a leading coefficient of seven and a constant term of four.
