Answer:
1. Negative leading coefficient, fifth degree
2. Positive leading coefficient, fourth degree
Step-by-step explanation:
The graph of a polynomial eventually rises or falls depending on its degree and the sign of the leading coefficient (lc = coefficient of term with the highest exponent)
[tex]\begin{array}{ccl}\textbf{Degree} & \textbf{Lc} & \textbf{End Behaviour}\\\text{Even} & + & \text{Up on left and right}\\\text{Even} & - & \text{Down on left; up on right}\\\text{Odd} & + & \text{Down on left and right}\\\text{Odd} & - & \text{Up on left; down on right}\\\end{array}[/tex]
In addition,
- the degree is at least the number of zeros and
- at least 1 greater than the number of local extrema ("bumps"), and
- a flattened zero or bump (flex point) shows that shows that it is degenerate (occurs multiple times)
Graph 1
Up on left, down on right — lc is negative; odd degree
Three zeros — at least third degree
Two bumps — at least third degree
Flex points — one at the origin: odd degree, so it occurs 3, 5, 7, or more times.
The polynomial has a positive leading coefficient and is probably fifth degree.
An example is the polynomial y = -x⁵ + x³ (Fig.1).
Graph 2
Up on left and right — lc is positive; even degree
Four zeros — at least fourth degree
Three bumps — at least fourth degree
Flex points — none obvious (although they could be present)
The polynomial has a positive leading coefficient and is probably fourth degree.
An example is the polynomial y = x⁴ - x³ - 4x² - 4x (Fig.2).
