Question

Algebra 2 please help

Answer 1

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)

$\begin{array}{ccl}\textbf{Degree} & \textbf{Lc} & \textbf{End Behaviour}\\\text{Even} & + & \text{Up on left and right}\\\text{Even} & - & \text{Down on left; up on right}\\\text{Odd} & + & \text{Down on left and right}\\\text{Odd} & - & \text{Up on left; down on right}\\\end{array}$

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).