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Precalculus: Mathematical for Calculus

James Stewart, Lothar Redlin, Saleem Watson

Chapter 3

Polynomial and Rational Functions - all with Video Answers

Educators

Section 1

Polynomial Functions and Their Graphs

02:05

Problem 1

Sketch the graph of each function by transforming the graph of an appropriate function of the form $y=x^{n}$ from Figure $2 .$ Indicate all $x$ - and $y$ -intercepts on each graph.
(a) $P(x)=x^{2}-4$
(b) $Q(x)=(x-4)^{2}$
(c) $R(x)=2 x^{2}-2$
(d) $S(x)=2(x-2)^{2}$

AG
Ankit Gupta
Numerade Educator
08:46

Problem 2

Sketch the graph of each function by transforming the graph of an appropriate function of the form $y=x^{n}$ from Figure $2 .$ Indicate all $x$ - and $y$ -intercepts on each graph.
(a) $P(x)=x^{4}-16$
(b) $Q(x)=(x+2)^{4}$
(c) $R(x)=(x+2)^{4}-16$
(d) $S(x)=-2(x+2)^{4}$

Peter Winans
Peter Winans
Numerade Educator
03:39

Problem 3

Sketch the graph of each function by transforming the graph of an appropriate function of the form $y=x^{n}$ from Figure $2 .$ Indicate all $x$ - and $y$ -intercepts on each graph.
(a) $P(x)=x^{3}-8$
(b) $Q(x)=-x^{3}+27$
(c) $R(x)=-(x+2)^{3}$
(d) $S(x)=\frac{1}{2}(x-1)^{3}+4$

AG
Ankit Gupta
Numerade Educator
04:07

Problem 4

Sketch the graph of each function by transforming the graph of an appropriate function of the form $y=x^{n}$ from Figure $2 .$ Indicate all $x$ - and $y$ -intercepts on each graph.
(a) $P(x)=(x+3)^{5}$
(b) $Q(x)=2(x+3)^{5}-64$
(c) $R(x)=-\frac{1}{2}(x-2)^{5}$
(d) $S(x)=-\frac{1}{2}(x-2)^{5}+16$

AG
Ankit Gupta
Numerade Educator
05:01

Problem 5

$5-10=$ Match the polynomial function with one of the graphs I-VI. Give reasons for your choice. (GRAPH CAN'T COPY)
$$P(x)=x\left(x^{2}-4\right)$$

Peter Winans
Peter Winans
Numerade Educator
04:21

Problem 6

$5-10=$ Match the polynomial function with one of the graphs I-VI. Give reasons for your choice. (GRAPH CAN'T COPY)
$$Q(x)=-x^{2}\left(x^{2}-4\right)$$

Peter Winans
Peter Winans
Numerade Educator
05:01

Problem 7

$5-10=$ Match the polynomial function with one of the graphs I-VI. Give reasons for your choice. (GRAPH CAN'T COPY)
$$R(x)=-x^{5}+5 x^{3}-4 x$$

Peter Winans
Peter Winans
Numerade Educator
05:01

Problem 8

$5-10=$ Match the polynomial function with one of the graphs I-VI. Give reasons for your choice. (GRAPH CAN'T COPY)
$$S(x)=\frac{1}{2} x^{6}-2 x^{4}$$

Peter Winans
Peter Winans
Numerade Educator
05:01

Problem 9

$5-10=$ Match the polynomial function with one of the graphs I-VI. Give reasons for your choice. (GRAPH CAN'T COPY)
$$T(x)=x^{4}+2 x^{3}$$

Peter Winans
Peter Winans
Numerade Educator
05:01

Problem 10

$5-10=$ Match the polynomial function with one of the graphs I-VI. Give reasons for your choice. (GRAPH CAN'T COPY)
$$U(x)=-x^{3}+2 x^{2}$$

Peter Winans
Peter Winans
Numerade Educator
00:54

Problem 11

Sketch the graph of the polynomial function. Make sure your graph shows all intercepts and exhibits the proper end behavior.
$$P(x)=(x-1)(x+2)$$

AG
Ankit Gupta
Numerade Educator
01:13

Problem 12

Sketch the graph of the polynomial function. Make sure your graph shows all intercepts and exhibits the proper end behavior.
$$P(x)=(x-1)(x+1)(x-2)$$

AG
Ankit Gupta
Numerade Educator
01:05

Problem 13

Sketch the graph of the polynomial function. Make sure your graph shows all intercepts and exhibits the proper end behavior.
$$P(x)=x(x-3)(x+2)$$

AG
Ankit Gupta
Numerade Educator
01:20

Problem 14

Sketch the graph of the polynomial function. Make sure your graph shows all intercepts and exhibits the proper end behavior.
$$P(x)=(2 x-1)(x+1)(x+3)$$

AG
Ankit Gupta
Numerade Educator
01:08

Problem 15

Sketch the graph of the polynomial function. Make sure your graph shows all intercepts and exhibits the proper end behavior.
$$P(x)=(x-3)(x+2)(3 x-2)$$

AG
Ankit Gupta
Numerade Educator
01:11

Problem 16

Sketch the graph of the polynomial function. Make sure your graph shows all intercepts and exhibits the proper end behavior.
$$P(x)=\frac{1}{5} x(x-5)^{2}$$

AG
Ankit Gupta
Numerade Educator
06:27

Problem 17

Sketch the graph of the polynomial function. Make sure your graph shows all intercepts and exhibits the proper end behavior.
$$P(x)=(x-1)^{2}(x-3)$$

Peter Winans
Peter Winans
Numerade Educator
05:57

Problem 18

Sketch the graph of the polynomial function. Make sure your graph shows all intercepts and exhibits the proper end behavior.
$$P(x)=\frac{1}{4}(x+1)^{3}(x-3)$$

Peter Winans
Peter Winans
Numerade Educator
01:49

Problem 19

Sketch the graph of the polynomial function. Make sure your graph shows all intercepts and exhibits the proper end behavior.
$$P(x)=\frac{1}{12}(x+2)^{2}(x-3)^{2}$$

AG
Ankit Gupta
Numerade Educator
01:33

Problem 20

Sketch the graph of the polynomial function. Make sure your graph shows all intercepts and exhibits the proper end behavior.
$$P(x)=(x-1)^{2}(x+2)^{3}$$

AG
Ankit Gupta
Numerade Educator
02:08

Problem 21

Sketch the graph of the polynomial function. Make sure your graph shows all intercepts and exhibits the proper end behavior.
$$P(x)=x^{3}(x+2)(x-3)^{2}$$

AG
Ankit Gupta
Numerade Educator
01:27

Problem 22

Sketch the graph of the polynomial function. Make sure your graph shows all intercepts and exhibits the proper end behavior.
$$P(x)=(x-3)^{2}(x+1)^{2}$$

AG
Ankit Gupta
Numerade Educator
01:03

Problem 23

Factor the polynomial and use the factored form to find the zeros. Then sketch the graph.
$$P(x)=x^{3}-x^{2}-6 x$$

AG
Ankit Gupta
Numerade Educator
00:53

Problem 24

Factor the polynomial and use the factored form to find the zeros. Then sketch the graph.
$$P(x)=x^{3}+2 x^{2}-8 x$$

AG
Ankit Gupta
Numerade Educator
01:04

Problem 25

Factor the polynomial and use the factored form to find the zeros. Then sketch the graph.
$$P(x)=-x^{3}+x^{2}+12 x$$

AG
Ankit Gupta
Numerade Educator
01:04

Problem 26

Factor the polynomial and use the factored form to find the zeros. Then sketch the graph.
$$P(x)=-2 x^{3}-x^{2}+x$$

AG
Ankit Gupta
Numerade Educator
01:46

Problem 27

Factor the polynomial and use the factored form to find the zeros. Then sketch the graph.
$$P(x)=x^{4}-3 x^{3}+2 x^{2}$$

AG
Ankit Gupta
Numerade Educator
01:21

Problem 28

Factor the polynomial and use the factored form to find the zeros. Then sketch the graph.
$$P(x)=x^{5}-9 x^{3}$$

AG
Ankit Gupta
Numerade Educator
01:17

Problem 29

Factor the polynomial and use the factored form to find the zeros. Then sketch the graph.
$$P(x)=x^{3}+x^{2}-x-1$$

AG
Ankit Gupta
Numerade Educator
01:31

Problem 30

Factor the polynomial and use the factored form to find the zeros. Then sketch the graph.
$$P(x)=x^{3}+3 x^{2}-4 x-12$$

AG
Ankit Gupta
Numerade Educator
01:33

Problem 31

Factor the polynomial and use the factored form to find the zeros. Then sketch the graph.
$$P(x)=2 x^{3}-x^{2}-18 x+9$$

AG
Ankit Gupta
Numerade Educator
02:36

Problem 32

Factor the polynomial and use the factored form to find the zeros. Then sketch the graph.
$$P(x)=\frac{1}{8}\left(2 x^{4}+3 x^{3}-16 x-24\right)^{2}$$

AG
Ankit Gupta
Numerade Educator
01:37

Problem 33

Factor the polynomial and use the factored form to find the zeros. Then sketch the graph.
$$P(x)=x^{4}-2 x^{3}-8 x+16$$

AG
Ankit Gupta
Numerade Educator
01:30

Problem 34

Factor the polynomial and use the factored form to find the zeros. Then sketch the graph.
$$P(x)=x^{4}-2 x^{3}+8 x-16$$

AG
Ankit Gupta
Numerade Educator
01:24

Problem 35

Factor the polynomial and use the factored form to find the zeros. Then sketch the graph.
$$P(x)=x^{4}-3 x^{2}-4$$

AG
Ankit Gupta
Numerade Educator
01:11

Problem 36

Factor the polynomial and use the factored form to find the zeros. Then sketch the graph.
$$P(x)=x^{6}-2 x^{3}+1$$

AG
Ankit Gupta
Numerade Educator
01:01

Problem 37

Determine the end behavior of $P$. Compare the graphs of $P$ and $Q$ on large and small vicwing rectangles, as in Example $3(b)$
$$P(x)=3 x^{3}-x^{2}+5 x+1 ; \quad Q(x)=3 x^{3}$$

AG
Ankit Gupta
Numerade Educator
01:31

Problem 38

Determine the end behavior of $P$. Compare the graphs of $P$ and $Q$ on large and small vicwing rectangles, as in Example $3(b)$
$$P(x)=-\frac{1}{8} x^{3}+\frac{1}{4} x^{2}+12 x ; \quad Q(x)=-\frac{1}{8} x^{3}$$

AG
Ankit Gupta
Numerade Educator
01:20

Problem 39

Determine the end behavior of $P$. Compare the graphs of $P$ and $Q$ on large and small vicwing rectangles, as in Example $3(b)$
$$P(x)=x^{4}-7 x^{2}+5 x+5 ; \quad Q(x)=x^{4}$$

AG
Ankit Gupta
Numerade Educator
01:31

Problem 40

Determine the end behavior of $P$. Compare the graphs of $P$ and $Q$ on large and small vicwing rectangles, as in Example $3(b)$
$$P(x)=-x^{5}+2 x^{2}+x ; \quad Q(x)=-x^{5}$$

AG
Ankit Gupta
Numerade Educator
01:47

Problem 41

Determine the end behavior of $P$. Compare the graphs of $P$ and $Q$ on large and small vicwing rectangles, as in Example $3(b)$
$$P(x)=x^{11}-9 x^{9} ; \quad Q(x)=x^{11}$$

AG
Ankit Gupta
Numerade Educator
01:39

Problem 42

Determine the end behavior of $P$. Compare the graphs of $P$ and $Q$ on large and small vicwing rectangles, as in Example $3(b)$
$$P(x)=2 x^{2}-x^{12} ; \quad Q(x)=-x^{12}$$

AG
Ankit Gupta
Numerade Educator
03:19

Problem 43

$43-46=$ The graph of a polynomial function is given. From the graph, find
(a) the $x$ - and $y$ -intercepts
(b) the coordinates of all local extrema
(GRAPH CAN'T COPY)
$$P(x)=-x^{2}+4 x$$

Daniel Nolan
Daniel Nolan
Numerade Educator
01:45

Problem 44

$43-46=$ The graph of a polynomial function is given. From the graph, find
(a) the $x$ - and $y$ -intercepts
(b) the coordinates of all local extrema
(GRAPH CAN'T COPY)
$$P(x)=\frac{2}{9} x^{3}-x^{2}$$

Zach Steedman
Zach Steedman
Numerade Educator
03:19

Problem 45

$43-46=$ The graph of a polynomial function is given. From the graph, find
(a) the $x$ - and $y$ -intercepts
(b) the coordinates of all local extrema
(GRAPH CAN'T COPY)
$$P(x)=-\frac{1}{2} x^{3}+\frac{3}{2} x-1$$

Daniel Nolan
Daniel Nolan
Numerade Educator
01:45

Problem 46

$43-46=$ The graph of a polynomial function is given. From the graph, find
(a) the $x$ - and $y$ -intercepts
(b) the coordinates of all local extrema
(GRAPH CAN'T COPY)
$$P(x)=\frac{1}{9} x^{4}-\frac{4}{9} x^{3}$$

Zach Steedman
Zach Steedman
Numerade Educator
01:14

Problem 47

Graph the polynomial in the given viewing rectangle. Find the coordinates of all local extrema. State each answer correct to two decimal places.
$$y=-x^{2}+8 x, \quad[-4,12] \text { by }[-50,30]$$

Peter Winans
Peter Winans
Numerade Educator
02:17

Problem 48

Graph the polynomial in the given viewing rectangle. Find the coordinates of all local extrema. State each answer correct to two decimal places.
$$y=x^{3}-3 x^{2}, \quad[-2,5] \text { by }[-10,10]$$

Peter Winans
Peter Winans
Numerade Educator
03:23

Problem 49

Graph the polynomial in the given viewing rectangle. Find the coordinates of all local extrema. State each answer correct to two decimal places.
$$y=x^{3}-12 x+9, \quad[-5,5] \text { by }[-30,30]$$

Peter Winans
Peter Winans
Numerade Educator
03:25

Problem 50

Graph the polynomial in the given viewing rectangle. Find the coordinates of all local extrema. State each answer correct to two decimal places.
$$y=2 x^{3}-3 x^{2}-12 x-32, \quad[-5,5] \text { by }[-60,30]$$

Peter Winans
Peter Winans
Numerade Educator
03:27

Problem 51

Graph the polynomial in the given viewing rectangle. Find the coordinates of all local extrema. State each answer correct to two decimal places.
$$y=x^{4}+4 x^{3}, \quad[-5,5] \text { by }[-30,30]$$

Peter Winans
Peter Winans
Numerade Educator
03:10

Problem 52

Graph the polynomial in the given viewing rectangle. Find the coordinates of all local extrema. State each answer correct to two decimal places.
$$y=x^{4}-18 x^{2}+32, \quad[-5,5] \text { by }[-100,100]$$

Peter Winans
Peter Winans
Numerade Educator
02:45

Problem 53

Graph the polynomial in the given viewing rectangle. Find the coordinates of all local extrema. State each answer correct to two decimal places.
$$y=3 x^{5}-5 x^{3}+3, \quad[-3,3] \text { by }[-5,10]$$

Peter Winans
Peter Winans
Numerade Educator
02:22

Problem 54

Graph the polynomial in the given viewing rectangle. Find the coordinates of all local extrema. State each answer correct to two decimal places.
$$y=x^{5}-5 x^{2}+6,[-3,3] \text { by }[-5,10]$$

Peter Winans
Peter Winans
Numerade Educator
01:31

Problem 55

Graph the polynomial and determine how many local maxima and minima it has.
$$y=-2 x^{2}+3 x+5$$

AG
Ankit Gupta
Numerade Educator
01:19

Problem 56

Graph the polynomial and determine how many local maxima and minima it has.
$$y=x^{3}+12 x$$

AG
Ankit Gupta
Numerade Educator
01:42

Problem 57

Graph the polynomial and determine how many local maxima and minima it has.
$$y=x^{3}-x^{2}-x$$

AG
Ankit Gupta
Numerade Educator
01:37

Problem 58

Graph the polynomial and determine how many local maxima and minima it has.
$$y=6 x^{3}+3 x+1$$

AG
Ankit Gupta
Numerade Educator
01:15

Problem 59

Graph the polynomial and determine how many local maxima and minima it has.
$$y=x^{4}-5 x^{2}+4$$

AG
Ankit Gupta
Numerade Educator
00:57

Problem 60

Graph the polynomial and determine how many local maxima and minima it has.
$$y=1.2 x^{5}+3.75 x^{4}-7 x^{3}-15 x^{2}+18 x$$

AG
Ankit Gupta
Numerade Educator
01:47

Problem 61

Graph the polynomial and determine how many local maxima and minima it has.
$$y=(x-2)^{5}+32$$

AG
Ankit Gupta
Numerade Educator
01:20

Problem 62

Graph the polynomial and determine how many local maxima and minima it has.
$$y=\left(x^{2}-2\right)^{3}$$

AG
Ankit Gupta
Numerade Educator
00:57

Problem 63

Graph the polynomial and determine how many local maxima and minima it has.
$$y=x^{8}-3 x^{4}+x$$

AG
Ankit Gupta
Numerade Educator
01:13

Problem 64

Graph the polynomial and determine how many local maxima and minima it has.
$$y=\frac{1}{3} x^{7}-17 x^{2}+7$$

AG
Ankit Gupta
Numerade Educator
02:16

Problem 65

Graph the family of polynomials in the same viewing rectangle, using the given values of $c .$ Explain how changing the value of $c$ affects the graph.
$$P(x)=c x^{3} ; \quad c=1,2,5, \frac{1}{2}$$

AG
Ankit Gupta
Numerade Educator
02:10

Problem 66

Graph the family of polynomials in the same viewing rectangle, using the given values of $c .$ Explain how changing the value of $c$ affects the graph.
$$P(x)=(x-c)^{4} ; \quad c=-1,0,1,2$$

AG
Ankit Gupta
Numerade Educator
01:48

Problem 67

Graph the family of polynomials in the same viewing rectangle, using the given values of $c .$ Explain how changing the value of $c$ affects the graph.
$$P(x)=x^{4}+c ; \quad c=-1,0,1,2$$

AG
Ankit Gupta
Numerade Educator
02:49

Problem 68

Graph the family of polynomials in the same viewing rectangle, using the given values of $c .$ Explain how changing the value of $c$ affects the graph.
$$P(x)=x^{3}+c x ; \quad c=2,0,-2,-4$$

AG
Ankit Gupta
Numerade Educator
02:54

Problem 69

Graph the family of polynomials in the same viewing rectangle, using the given values of $c .$ Explain how changing the value of $c$ affects the graph.
$$P(x)=x^{4}-c x ; \quad c=0,1,8,27$$

AG
Ankit Gupta
Numerade Educator
02:14

Problem 70

Graph the family of polynomials in the same viewing rectangle, using the given values of $c .$ Explain how changing the value of $c$ affects the graph.
$$P(x)=x^{c} ; \quad c=1,3,5,7$$

AG
Ankit Gupta
Numerade Educator
05:41

Problem 71

(a) On the same coordinate axes, sketch graphs (as accurately as possible) of the functions $y=x^{3}-2 x^{2}-x+2 \quad$ and $\quad y=-x^{2}+5 x+2$
(b) Based on your sketch in part (a), at how many points do the two graphs appear to intersect?
(c) Find the coordinates of all intersection points.

AG
Ankit Gupta
Numerade Educator
02:35

Problem 72

Portions of the graphs of $y=x^{2}, y=x^{3}, y=x^{4}, y=x^{5},$ and $y=x^{6}$ are plotted in the figures. Determine which function belongs to each graph. (GRAPH CAN'T COPY)

Zach Steedman
Zach Steedman
Numerade Educator
06:17

Problem 73

Recall that a function $f$ is odd if $f(-x)=-f(x)$ or even if $f(-x)=f(x)$ for all real $x$
(a) Show that a polynomial $P(x)$ that contains only odd powers of $x$ is an odd function.
(b) Show that a polynomial $P(x)$ that contains only even powers of $x$ is an even function.
(c) Show that if a polynomial $P(x)$ contains both odd and cven powers of $x,$ then it is neither an odd nor an even function.
(d) Express the function
$$P(x)=x^{5}+6 x^{3}-x^{2}-2 x+5$$ as the sum of an odd function and an even function.

Peter Winans
Peter Winans
Numerade Educator
04:58

Problem 74

(a) Graph the function $P(x)=(x-1)(x-3)(x-4)$ and find all local extrema, correct to the nearest tenth.
(b) Graph the function
$$Q(x)=(x-1)(x-3)(x-4)+5$$
and use your answers to part (a) to find all local extrema, correct to the nearest tenth.

Peter Winans
Peter Winans
Numerade Educator
03:38

Problem 75

(a) Graph the function $P(x)=(x-2)(x-4)(x-5)$ and determine how many local extrema it has.
(b) If $a<b<c,$ explain why the function
$$P(x)=(x-a)(x-b)(x-c)$$ must have two local extrema.

Peter Winans
Peter Winans
Numerade Educator
04:14

Problem 76

(a) How many $x$ -intercepts and how many local extrema does the polynomial $P(x)=x^{3}-4 x$ have?
(b) How many $x$ -intercepts and how many local extrema does the polynomial $Q(x)=x^{3}+4 x$ have?
(c) If $a>0,$ how many $x$ -intercepts and how many local extrema does each of the polynomials $P(x)=x^{3}-a x$ and $Q(x)=x^{3}+a x$ have? Explain your answer.

AG
Ankit Gupta
Numerade Educator
05:01

Problem 77

Market Research A market analyst working for a smallappliance manufacturer finds that if the firm produces and sells $x$ blenders annually, the total profit (in dollars) is
$$
P(x)=8 x+0.3 x^{2}-0.0013 x^{3}-372
$$
Graph the function $P$ in an appropriate viewing rectangle and use the graph to answer the following questions.
(a) When just a few blenders are manufactured, the firm loses money (profit is negative). (For example, $P(10)=-263.3,$ so the firm loses $\$ 263.30$ if it produces and sells only 10 blenders.) How many blenders must the firm produce to break even?
(b) Does profit increase indefinitely as more blenders are produced and sold? If not, what is the largest possible profit the firm could have?

Sheryl Ezze
Sheryl Ezze
Numerade Educator
02:12

Problem 78

Population Change The rabbit population on a small island is observed to be given by the function
$$P(t)=120 t-0.4 t^{4}+1000$$
where $t$ is the time (in months) since observations of the island began.
(a) When is the maximum population attained, and what is that maximum population?
(b) When does the rabbit population disappear from the island?
(GRAPH CAN'T COPY)

Zach Steedman
Zach Steedman
Numerade Educator
05:09

Problem 79

Volume of a Box An open box is to be constructed from
a picce of cardboard $20 \mathrm{cm}$ by $40 \mathrm{cm}$ by cutting squares of side length $x$ from each corner and folding up the sides, as shown in the figure.
(a) Express the volume $V$ of the box as a function of $x .$
(b) What is the domain of $V ?$ (Use the fact that length and volume must be positive.)
(c) Draw a graph of the function $V$ and use it to estimate the maximum volume for such a box.
(FIGURE CAN'T COPY)

Peter Winans
Peter Winans
Numerade Educator
05:55

Problem 80

Volume of a Box $A$ cardboard box has a square base, with each edge of the base having length $x$ inches, as shown in the figure. The total length of all 12 edges of the box is
144 in.
(a) Show that the volume of the box is given by the function $V(x)=2 x^{2}(18-x)$
(b) What is the domain of $V ?$ (Use the fact that length and volume must be positive.)
(c) Draw a graph of the function $V$ and use it to estimate the maximum volume for such a box.
(FIGURE CAN'T COPY)

Zach Steedman
Zach Steedman
Numerade Educator
04:43

Problem 81

Graphs of Large Powers Graph the functions $y=x^{2}$ $y=x^{3}, y=x^{4},$ and $y=x^{5},$ for $-1 \leq x \leq 1,$ on the same coordinate axes. What do you think the graph of $y=x^{100}$ would look like on this same interval? What about $y=x^{101} ?$ Make a table of values to confirm your answers.

Peter Winans
Peter Winans
Numerade Educator
01:21

Problem 82

Maximum Number of Local Extrema What is the smallest possible degree that the polynomial whose graph is shown can have? Explain.
(FIGURE CAN'T COPY)

Peter Winans
Peter Winans
Numerade Educator
04:28

Problem 83

Possible Number of Local Extrema Is it possible for a third-degree polynomial to have exactly one local extremum? Can a fourth-degree polynomial have exactly two local extrema? How many local extrema can polynomials of third, fourth, fifth, and sixth degree have? (Think about the end behavior of such polynomials.) Now give an example of a polynomial that has six local extrema.

AG
Ankit Gupta
Numerade Educator
00:46

Problem 84

Impossible Situation? Is it possible for a polynomial to have two local maxima and no local minimum? Explain.

Zach Steedman
Zach Steedman
Numerade Educator