Question

write the polynomial in standard form, find the degree and leading coefficient of the
polynomial then classify the polynomial by the number of turns
1: 6c^2+2c^4-c
2: 7+3p^2
3: 3t^8
4: 3.14159r^2 - 5over7r^8+2r^5
5: 4w^11-w^12
6: 8d-2-4d^3
7: 5z+2z^3+3z^4

Answer 1

For each polynomial below I put it in standard form (descending powers), give the degree, the leading coefficient, and the maximum possible number of turns (turns = possible local extrema; maximum = degree − 1).

1) 6c^2 + 2c^4 − c
- Standard form: 2c^4 + 6c^2 − c
- Degree: 4
- Leading coefficient: 2
- Maximum number of turns: 3

2) 7 + 3p^2
- Standard form: 3p^2 + 7
- Degree: 2
- Leading coefficient: 3
- Maximum number of turns: 1

3) 3t^8
- Standard form: 3t^8
- Degree: 8
- Leading coefficient: 3
- Maximum number of turns: 7

4) 3.14159 r^2 − (5/7) r^8 + 2 r^5
- Standard form: −(5/7) r^8 + 2 r^5 + 3.14159 r^2
- Degree: 8
- Leading coefficient: −5/7
- Maximum number of turns: 7

5) 4w^11 − w^12
- Standard form: −w^12 + 4w^11
- Degree: 12
- Leading coefficient: −1
- Maximum number of turns: 11

6) 8d − 2 − 4d^3
- Standard form: −4d^3 + 8d − 2
- Degree: 3
- Leading coefficient: −4
- Maximum number of turns: 2

7) 5z + 2z^3 + 3z^4
- Standard form: 3z^4 + 2z^3 + 5z
- Degree: 4
- Leading coefficient: 3
- Maximum number of turns: 3

Note: the listed number of turns is the maximum possible (a polynomial of degree n can have at most n−1 turning points); the actual graph may have fewer.