[Video] Behavior of Polynomial Graphs

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In this set of videos, we will work to understand how polynomials behave and how their graphs look.

Question 1: What is the multiplicity of a polynomial at a root?

(Click the arrow to the right to see the first question.)

(Click the arrow to the right to see the question posed at the end of the video.)

The function $f(x) = 17x^4(x+1)^3(x-1)^2(x-2)$ has multiplicity what at root $x=1$ ?

$0$ $1$ $2$ $3$ $4$

(Click the arrow to the right to see an example.)

Example 1

Question 2: Matching polynomial functions to their graphs.

(Click the arrow to the right to see the second question.)

(Click the arrow to the right to see the question posed at the end of the video.)

Which of the following graphs is the graph of the function $f(x) = \frac {1}{6}x^2(x-2)^2(x-1)(x+1)^2(x+2)$ ?

(Click the arrow to the right to see an example.)

Example 2

Question 3: Matching polynomial graphs to their functions.

(Click the arrow to the right to see the third question.)

(Click the arrow to the right to see the question posed at the end of the video.)

Which of the following functions is the graph below?

$\frac {1}{7}x^2(x-2)(x-1)(x+1)^2(x+2)$ $-\frac {1}{7}x^2(x-2)(x-1)(x+1)^2(x+2)$ $\frac {1}{7}x(x-2)^2(x-1)^2(x+1)(x+2)^2$ $-\frac {1}{7}x(x-2)^2(x-1)^2(x+1)(x+2)^2$ $\frac {1}{7}x(x-2)^2(x-1)(x+1)^2(x+2)^2$ $-\frac {1}{7}x(x-2)^2(x-1)(x+1)^2(x+2)^2$

(Click the arrow to the right to see an example.)

Example 3

Question 4: Factoring polynomials using their graphs.

(You may want to review factoring polynomials before working on question 4. Once you are prepared to work on question 4, click the arrow to the right.)

(Click the arrow to the right to see the question posed at the end of the video.)

Given the graph of the function $f(x) = \frac {1}{5}x^5 + \frac {1}{5}x^4 - 2x^3 - \frac {4}{5}x^2 + \frac {24}{5}x$ below, factor $f(x)$

$\frac {4}{5}x(x-2)(x+2)(x+3)$ $\frac {1}{5}x(x-2)(x+2)^2(x+3)$ $\frac {1}{5}x(x-2)^2(x+2)(x+3)$ $\frac {1}{5}x(x-2)(x+2)(x+3)$ $\frac {4}{5}x(x-2)^2(x+2)(x+3)$ $\frac {1}{5}x^3(x-2)^4(x+2)^3(x+3)$

(Click the arrow to the right to see an example.)

Example 4