Answer :
Sure! Let's go through each polynomial step-by-step to understand their structure and components.
1. Polynomial: [tex]\(14a + 21a^2 + 21a^3\)[/tex]
- This expression is a polynomial in [tex]\(a\)[/tex].
- It starts with a linear term [tex]\(14a\)[/tex], followed by a quadratic term [tex]\(21a^2\)[/tex], and a cubic term [tex]\(21a^3\)[/tex].
- The polynomial is already in its simplest form.
2. Polynomial: [tex]\(10n^3 - 9n^2 + n\)[/tex]
- This is a polynomial in [tex]\(n\)[/tex].
- It includes a cubic term [tex]\(10n^3\)[/tex], a quadratic term [tex]\(-9n^2\)[/tex], and a linear term [tex]\(n\)[/tex].
- The polynomial is already simplified.
3. Polynomial: [tex]\(-28v^2 - 8v - 36\)[/tex]
- This is a polynomial in [tex]\(v\)[/tex].
- It consists of a quadratic term [tex]\(-28v^2\)[/tex], a linear term [tex]\(-8v\)[/tex], and a constant term [tex]\(-36\)[/tex].
- The polynomial is simplified as given.
4. Polynomial: [tex]\(-8x^7 + 24x^6 + 12x^5\)[/tex]
- This expression is a polynomial in [tex]\(x\)[/tex].
- It contains a 7th-degree term [tex]\(-8x^7\)[/tex], a 6th-degree term [tex]\(24x^6\)[/tex], and a 5th-degree term [tex]\(12x^5\)[/tex].
- The polynomial is provided in its simplest form.
5. Polynomial: [tex]\(20 - 35n^2 - 20n^3\)[/tex]
- This is a polynomial in [tex]\(n\)[/tex].
- It includes a constant term [tex]\(20\)[/tex], a quadratic term [tex]\(-35n^2\)[/tex], and a cubic term [tex]\(-20n^3\)[/tex].
- The expression is already simplified.
6. Polynomial: [tex]\(9x^6 - 63x^3 - 90x^2\)[/tex]
- This polynomial is in terms of [tex]\(x\)[/tex].
- It starts with a 6th-degree term [tex]\(9x^6\)[/tex], followed by a cubic term [tex]\(-63x^3\)[/tex], and a quadratic term [tex]\(-90x^2\)[/tex].
- This polynomial is in its simplest form.
7. Polynomial: [tex]\(-3k^3 + 15k^2 - 6k\)[/tex]
- This expression is a polynomial in [tex]\(k\)[/tex].
- It has a cubic term [tex]\(-3k^3\)[/tex], a quadratic term [tex]\(15k^2\)[/tex], and a linear term [tex]\(-6k\)[/tex].
- The polynomial is as simple as it gets.
8. Polynomial: [tex]\(50p^3 + 50p^2 - 20\)[/tex]
- This is a polynomial in [tex]\(p\)[/tex].
- It consists of a cubic term [tex]\(50p^3\)[/tex], a quadratic term [tex]\(50p^2\)[/tex], and a constant term [tex]\(-20\)[/tex].
- The expression is already in a simplified form.
9. Polynomial: [tex]\(32n^3 + 28n - 20\)[/tex]
- This polynomial is in terms of [tex]\(n\)[/tex].
- It includes a cubic term [tex]\(32n^3\)[/tex], a linear term [tex]\(28n\)[/tex], and a constant term [tex]\(-20\)[/tex].
- It's presented in its simplest form.
10. Polynomial: [tex]\(-90x^5 + 100x + 60\)[/tex]
- This expression is a polynomial in [tex]\(x\)[/tex].
- It has a 5th-degree term [tex]\(-90x^5\)[/tex], a linear term [tex]\(100x\)[/tex], and a constant term [tex]\(60\)[/tex].
- The polynomial is in its simplest form.
11. Polynomial: [tex]\(3m^2 + 9m + 27\)[/tex]
- This is a polynomial in [tex]\(m\)[/tex].
- It includes a quadratic term [tex]\(3m^2\)[/tex], a linear term [tex]\(9m\)[/tex], and a constant term [tex]\(27\)[/tex].
- The polynomial is already simplified.
12. Polynomial: [tex]\(12r^2 + 4r - 12\)[/tex]
- This polynomial is in terms of [tex]\(r\)[/tex].
- It consists of a quadratic term [tex]\(12r^2\)[/tex], a linear term [tex]\(4r\)[/tex], and a constant term [tex]\(-12\)[/tex].
- The expression is in its simplest form.
13. Polynomial: [tex]\(64 + 40x^2 + 72x\)[/tex]
- This is a polynomial in [tex]\(x\)[/tex].
- It includes a constant term [tex]\(64\)[/tex], a quadratic term [tex]\(40x^2\)[/tex], and a linear term [tex]\(72x\)[/tex].
- The polynomial is provided in a simplified form.
14. Polynomial: [tex]\(-18n^2 + 15n - 15\)[/tex]
- This polynomial is in terms of [tex]\(n\)[/tex].
- It has a quadratic term [tex]\(-18n^2\)[/tex], a linear term [tex]\(15n\)[/tex], and a constant term [tex]\(-15\)[/tex].
- The expression is already simplified.
These are the details of each polynomial based on their form and degree. Each one is expressed in a standard and simplified polynomial form, comprising variable terms and constants.
1. Polynomial: [tex]\(14a + 21a^2 + 21a^3\)[/tex]
- This expression is a polynomial in [tex]\(a\)[/tex].
- It starts with a linear term [tex]\(14a\)[/tex], followed by a quadratic term [tex]\(21a^2\)[/tex], and a cubic term [tex]\(21a^3\)[/tex].
- The polynomial is already in its simplest form.
2. Polynomial: [tex]\(10n^3 - 9n^2 + n\)[/tex]
- This is a polynomial in [tex]\(n\)[/tex].
- It includes a cubic term [tex]\(10n^3\)[/tex], a quadratic term [tex]\(-9n^2\)[/tex], and a linear term [tex]\(n\)[/tex].
- The polynomial is already simplified.
3. Polynomial: [tex]\(-28v^2 - 8v - 36\)[/tex]
- This is a polynomial in [tex]\(v\)[/tex].
- It consists of a quadratic term [tex]\(-28v^2\)[/tex], a linear term [tex]\(-8v\)[/tex], and a constant term [tex]\(-36\)[/tex].
- The polynomial is simplified as given.
4. Polynomial: [tex]\(-8x^7 + 24x^6 + 12x^5\)[/tex]
- This expression is a polynomial in [tex]\(x\)[/tex].
- It contains a 7th-degree term [tex]\(-8x^7\)[/tex], a 6th-degree term [tex]\(24x^6\)[/tex], and a 5th-degree term [tex]\(12x^5\)[/tex].
- The polynomial is provided in its simplest form.
5. Polynomial: [tex]\(20 - 35n^2 - 20n^3\)[/tex]
- This is a polynomial in [tex]\(n\)[/tex].
- It includes a constant term [tex]\(20\)[/tex], a quadratic term [tex]\(-35n^2\)[/tex], and a cubic term [tex]\(-20n^3\)[/tex].
- The expression is already simplified.
6. Polynomial: [tex]\(9x^6 - 63x^3 - 90x^2\)[/tex]
- This polynomial is in terms of [tex]\(x\)[/tex].
- It starts with a 6th-degree term [tex]\(9x^6\)[/tex], followed by a cubic term [tex]\(-63x^3\)[/tex], and a quadratic term [tex]\(-90x^2\)[/tex].
- This polynomial is in its simplest form.
7. Polynomial: [tex]\(-3k^3 + 15k^2 - 6k\)[/tex]
- This expression is a polynomial in [tex]\(k\)[/tex].
- It has a cubic term [tex]\(-3k^3\)[/tex], a quadratic term [tex]\(15k^2\)[/tex], and a linear term [tex]\(-6k\)[/tex].
- The polynomial is as simple as it gets.
8. Polynomial: [tex]\(50p^3 + 50p^2 - 20\)[/tex]
- This is a polynomial in [tex]\(p\)[/tex].
- It consists of a cubic term [tex]\(50p^3\)[/tex], a quadratic term [tex]\(50p^2\)[/tex], and a constant term [tex]\(-20\)[/tex].
- The expression is already in a simplified form.
9. Polynomial: [tex]\(32n^3 + 28n - 20\)[/tex]
- This polynomial is in terms of [tex]\(n\)[/tex].
- It includes a cubic term [tex]\(32n^3\)[/tex], a linear term [tex]\(28n\)[/tex], and a constant term [tex]\(-20\)[/tex].
- It's presented in its simplest form.
10. Polynomial: [tex]\(-90x^5 + 100x + 60\)[/tex]
- This expression is a polynomial in [tex]\(x\)[/tex].
- It has a 5th-degree term [tex]\(-90x^5\)[/tex], a linear term [tex]\(100x\)[/tex], and a constant term [tex]\(60\)[/tex].
- The polynomial is in its simplest form.
11. Polynomial: [tex]\(3m^2 + 9m + 27\)[/tex]
- This is a polynomial in [tex]\(m\)[/tex].
- It includes a quadratic term [tex]\(3m^2\)[/tex], a linear term [tex]\(9m\)[/tex], and a constant term [tex]\(27\)[/tex].
- The polynomial is already simplified.
12. Polynomial: [tex]\(12r^2 + 4r - 12\)[/tex]
- This polynomial is in terms of [tex]\(r\)[/tex].
- It consists of a quadratic term [tex]\(12r^2\)[/tex], a linear term [tex]\(4r\)[/tex], and a constant term [tex]\(-12\)[/tex].
- The expression is in its simplest form.
13. Polynomial: [tex]\(64 + 40x^2 + 72x\)[/tex]
- This is a polynomial in [tex]\(x\)[/tex].
- It includes a constant term [tex]\(64\)[/tex], a quadratic term [tex]\(40x^2\)[/tex], and a linear term [tex]\(72x\)[/tex].
- The polynomial is provided in a simplified form.
14. Polynomial: [tex]\(-18n^2 + 15n - 15\)[/tex]
- This polynomial is in terms of [tex]\(n\)[/tex].
- It has a quadratic term [tex]\(-18n^2\)[/tex], a linear term [tex]\(15n\)[/tex], and a constant term [tex]\(-15\)[/tex].
- The expression is already simplified.
These are the details of each polynomial based on their form and degree. Each one is expressed in a standard and simplified polynomial form, comprising variable terms and constants.
