# Understanding Polynomial Functions - Video Tutorials & Practice Problems

### Introduction to Polynomial Functions

Determine if the given function is a polynomial function. If so, write in standard form, then state the degree and leading coefficient. $f\left(x\right)=4x^3+\frac12x^{-1}-2x+1$

Polynomial with $n=3,a_{n}=4$

Polynomial with $n=4,a_{n}=3$

Polynomial with $n=-1,a_{n}=\frac12$

Not a polynomial function.

Determine if the given function is a polynomial function. If so, write in standard form, then state the degree and leading coefficient. $f\left(x\right)=2+x$

Polynomial with $n=1,a_{n}=2$

Polynomial with $n=0,a_{n}=1$

Polynomial with $n=1,a_{n}=1$

Not a polynomial function.

Determine if the given function is a polynomial function. If so, write in standard form, then state the degree and leading coefficient. $f\left(x\right)=3x^2+5x+2$

Polynomial with $n=3,a_{n}=2$

Polynomial with $n=2,a_{n}=3$

Polynomial with $n=2,a_{n}=2$

Not a polynomial function.

### End Behavior of Polynomial Functions

Determine the end behavior of the given polynomial function. $f\left(x\right)=x^2+4x+x+7x^3$

Right side rises; Ends are same

Right side rises; Ends are opposite

Right side falls; Ends are same

Right side falls; Ends are opposite

Match the given polynomial function to its graph based on end behavior. $f\left(x\right)=-2x^3+x^2+1$

### Finding Zeros & Their Multiplicity

Find the zeros of the given polynomial function and give the multiplicity of each. State whether the graph crosses or touches the x-axis at each zero. $f\left(x\right)=2x^4-12x^3+18x^2$

Touch at $x=0$, Cross at $x=-3$

Touch at $x=0$, Touch at $x=3$

Touch at $x=1,$ Cross at $x=-3$

Touch at $x=-1$, Cross at $x=0$

Find the zeros of the given polynomial function and give the multiplicity of each. State whether the graph crosses or touches the x-axis at each zero. $f\left(x\right)=x^2\left(x-1\right)^3\left(2x+6\right)$

Cross at $x=0,$ Cross at $x=1,$ Cross at $x=3$

Touch at $x=0,$ Cross at $x=-1,$ Cross at $x=3$

Cross at $x=0,$ Touch at $x=1,$ Touch at $x=-3$

Touch at $x=0,$ Cross at $x=1,$ Cross at $x=-3$

### Maximum Turning Points of a Polynomial Function

Determine the maximum number of turning points for the given polynomial function. $f\left(x\right)=6x^4+2x$

1

2

3

4

Based *ONLY* on the maximum number of turning points, which of the following graphs could NOT be the graph of the given function? $f\left(x\right)=x^3+1$

The given term represents the leading term of some polynomial function. Determine the end behavior and the maximum number of turning points. $4x^5$

Right side rises; Ends are opposite & 4 maximum turning points

Right side rises; Ends are opposite & 5 maximum turning points

Right side rises; Ends are the same & 4 maximum turning points

Right side falls; Ends are opposite & 4 maximum turning points