# 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=$n=2,a_{n}=2$

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

## Do you want more practice?

- In Exercises 1–10, determine which functions are polynomial functions. For those that are, identify the degree...
- In Exercises 1–10, determine which functions are polynomial functions. For those that are, identify the degree...
- In Exercises 1–10, determine which functions are polynomial functions. For those that are, identify the degree...
- Graph each function. Determine the largest open intervals of the domain over which each function is (a) increa...
- In Exercises 10–13, use the Leading Coefficient Test to determine the end behavior of the graph of the given p...
- In Exercises 10–13, use the Leading Coefficient Test to determine the end behavior of the graph of the given p...
- In Exercises 11–14, identify which graphs are not those of polynomial functions.
- In Exercises 11–14, identify which graphs are not those of polynomial functions.
- Graph each function. Determine the largest open intervals of the domain over which each function is (a) increa...
- Graph the following on the same coordinate system. (a) y = x^2 (b) y = 3x^2 (c) y = 1/3x^2 (d) How does the c...
- In Exercises 19–24, (a) Use the Leading Coefficient Test to determine the graph's end behavior. (b) Determine...
- In Exercises 19–24, use the Leading Coefficient Test to determine the end behavior of the graph of the polynom...
- Graph each function. Determine the largest open intervals of the domain over which each function is (a) increa...
- In Exercises 19–24, (a) Use the Leading Coefficient Test to determine the graph's end behavior. (b) Determine...
- Use an end behavior diagram, , , , or , to describe the end behavior of the graph of each polynomial functi...
- Use an end behavior diagram, , , , or , to describe the end behavior of the graph of each polynomial functi...
- In Exercises 19–24, use the Leading Coefficient Test to determine the end behavior of the graph of the polynom...
- In Exercises 19–24, use the Leading Coefficient Test to determine the end behavior of the graph of the polynom...
- Use an end behavior diagram, , , , or , to describe the end behavior of the graph of each polynomial functi...
- In Exercises 25–26, graph each polynomial function. f(x) = 2x^2(x - 1)^3(x + 2)
- In Exercises 25–32, find the zeros for each polynomial function and give the multiplicity for each zero. State...
- In Exercises 25–32, find the zeros for each polynomial function and give the multiplicity for each zero. State...
- In Exercises 25–26, graph each polynomial function. f(x) = -x^3(x + 4)^2(x-1)
- In Exercises 25–32, find the zeros for each polynomial function and give the multiplicity for each zero. State...
- Graph each polynomial function. Factor first if the polynomial is not in factored form. See Examples 3 and 4. ...
- Graph each polynomial function. Factor first if the polynomial is not in factored form. See Examples 3 and 4. ...
- In Exercises 33–40, use the Intermediate Value Theorem to show that each polynomial has a real zero between th...
- In Exercises 33–40, use the Intermediate Value Theorem to show that each polynomial has a real zero between th...
- Graph each polynomial function. Factor first if the polynomial is not in factored form. See Examples 3 and 4. ...
- In Exercises 33–40, use the Intermediate Value Theorem to show that each polynomial has a real zero between th...
- Determine the largest open interval of the domain (a) over which the function is increasing and (b) over which...
- Determine the largest open interval of the domain (a) over which the function is increasing and (b) over which...
- Determine the largest open interval of the domain (a) over which the function is increasing and (b) over which...
- If the given term is the dominating term of a polynomial function, what can we conclude about each of the foll...
- If the given term is the dominating term of a polynomial function, what can we conclude about each of the foll...
- Graph each polynomial function. ƒ(x)=(x-2)^2(x+3)
- Use the intermediate value theorem to show that each polynomial function has a real zero between the numbers g...
- Graph each polynomial function. ƒ(x)=2x^3+x^2-x
- Use the intermediate value theorem to show that each polynomial function has a real zero between the numbers g...
- Use the intermediate value theorem to show that each polynomial function has a real zero between the numbers g...
- Graph each polynomial function. ƒ(x)=-2x^4+7x^3-4x^2-4x
- For each polynomial function, identify its graph from choices A–F. ƒ(x)=(x-2)^2(x-5)
- For each polynomial function, identify its graph from choices A–F. ƒ(x)=-(x-2)^2(x-5)
- For each polynomial function, identify its graph from choices A–F. ƒ(x)=(x-2)^2(x-5)^2
- For each polynomial function, identify its graph from choices A–F. ƒ(x)=(x-2)(x-5)
- Show that the real zeros of each polynomial function satisfy the given conditions. See Example 6. ƒ(x)=x^4-x^3...
- For each polynomial function, identify its graph from choices A–F. ƒ(x)=-(x-2)(x-5)
- Show that the real zeros of each polynomial function satisfy the given conditions. See Example 6. ƒ(x)=2x^5-x^...
- For each polynomial function, identify its graph from choices A–F. ƒ(x)=-(x-2)^2(x-5)^2
- Show that the real zeros of each polynomial function satisfy the given conditions. See Example 6. ƒ(x)=x^4+x^3...
- Show that the real zeros of each polynomial function satisfy the given conditions. See Example 6. ƒ(x)=x^5+2x^...
- Show that the real zeros of each polynomial function satisfy the given conditions. See Example 6. ƒ(x)=3x^4+2x...
- Show that the real zeros of each polynomial function satisfy the given conditions. See Example 6. ƒ(x)=3x^4+2x...
- Show that the real zeros of each polynomial function satisfy the given conditions. See Example 6. ƒ(x)=x^5-3x^...
- Show that the real zeros of each polynomial function satisfy the given conditions. See Example 6. ƒ(x)=x^5-3x^...
- Find a polynomial function f of least degree having the graph shown. (Hint: See the NOTE following Example 4.)...
- Find a polynomial function f of least degree having the graph shown. (Hint: See the NOTE following Example 4.)...
- Use a graphing calculator to find the coordinates of the turning points of the graph of each polynomial functi...
- Use a graphing calculator to find the coordinates of the turning points of the graph of each polynomial functi...
- Use a graphing calculator to find the coordinates of the turning points of the graph of each polynomial functi...
- The following exercises are geometric in nature and lead to polynomial models. Solve each problem. A standard ...
- The following exercises are geometric in nature and lead to polynomial models. Solve each problem. A standard ...
- Exercises 107–109 will help you prepare for the material covered in the next section. Factor: x^3+3x^2−x−3
- Exercises 107–109 will help you prepare for the material covered in the next section. Determine whether f(x)=x...
- Rewrite 4-5x-x^2+6x^3 in descending powers of x.
- Use (2x^3−3x^2−11x+6)/(x−3)=2x^2+3x−2 to factor 2x^3-3x^2-11x+6 completely.