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Algebra 2 please help

Algebra 2 please help-example-1
User Fracca
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1 Answer

4 votes

Answer:

1. Negative leading coefficient, fifth degree

2. Positive leading coefficient, fourth degree

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).

Algebra 2 please help-example-1
Algebra 2 please help-example-2
User Imaliazhar
by
6.1k points