If the function is an even function, its graph is symmetric with respect to the y-axis, that is, f(–x) = f(x). For large positive or negative values of x, 17/ (8 x + 4) approaches zero, and the graph approximates the line y = (1/2) x - (7/4). How to find the Equation of a Polynomial Function from its Graph, How to find the Formula for a Polynomial Given: Zeros/Roots, Degree, and One Point, examples and step by step solutions, Find an Equation of a Degree 4 or 5 Polynomial Function From the Graph of the Function, PreCalculus A point in this system has two coordinates. 66 0 obj <>stream {'�_1�����s\���+H�w u�].��E�!� !�"�C%Y�%�N���%���B��r >e��u��\sw���,���2�������fW,S�7χ.S_��� ��b�l(ƈ��A�0�d�jve&�Yl=��]1��{� 29Hy��,u Q|]��a{%�� Zeros of the function f(x) are 0 and -2, and zeros of the function $ g(x)$ are 0 and 2. These cookies do not store any personal information. Notice in the case of the graph opens up to the right and down to the left. h�bbd```b``z"@$�ɶ,"� 9T@$�˲J�Hv0;�lk��+ˊ�H���t �h�b+f�Ȗ�`5� ��l�$ ��l5�ms��a`t�&�� �� Graph $ f(x) = x^4 – 4x^2 + x – 1$. \begin {aligned} f (x)&= (3x-2) (x+2)^2 \\\\ \tealD 0&= (3x-2) (x+2)^2\\ \\ \end {aligned} f (x) 0. . Any cookies that may not be particularly necessary for the website to function and is used specifically to collect user personal data via analytics, ads, other embedded contents are termed as non-necessary cookies. Given the graph of a step function, find the function's outputs for given specific inputs. This means that the ends of our graph will either decrease or increase without bound. If k > 1 the graph will flatten at $ x_0$. Example 3. Determine the y y -intercept, (0,P (0)) (0, P (0)). . If $ a < 0$ and n is odd the graph will decrease at the right end and increase at the left end. Math video on how to graph a factored polynomial function that is cubic (3rd degree). Besides predicting the end behavior of a function, it is possible to sketch a function, provided that you know its roots. (The main difference is how you treat a… Find the zeros of a polynomial function. Graph will intersect y – axis in (0, 8). �?�I�D�NB�*�K�p��p��/��ֈ�Hl 9��-��A�v���������� �!�����ﺗ,jg,*;�\S������ \�RO�}���և�'"VӼ�o�k'�i�K��z����4����� ������Y��곯l(G$���!��1��)����K��e���N��wtv�9̰���L��Z6F�N3��Y�:�ծ:?߬6��n�Q��PՍߙ�E� vL�M��ͧ����"����Ny#�.�� �M������_o������]�+v�e^XN ����&�2���w�Q=m�Yn�%� oMcV��=,��1� q�g -intercepts, we can solve the equation. Process for Graphing a Polynomial Determine all the zeroes of the polynomial and their multiplicity. Thus, a polynomial function p(x) has the following general form: Every polynomial function is continuous. y9��x���S��F�y�5H6d�����Rg@��Ƒ�u��k�$��C��w���Y"��0G�\S��(��N�8f�{z�z�H��'� N�h$ ���l�rhIFt=O���B),�T�T���8f�t��ꈳ��yMy�كy�¶3�N!��CT-�k�5}� 5�49��V�#������?npM�Рa��Z�� �|�gưЏ 3���Z݈T�J� 3:JC�5����H�V�1���+�!%���,��8jM���R�w��!���U1K2چU�����^τlI]O�:dc�d�����:�D���1x��A�W�)���.�bo��1֫���/�x�e�ঘ�>� T�!07X��4뫬�pRh��#�h�ZӅ�{��֝w� �{���J/�y�)q0X�H��{��O����~�:�6{���x���k��5�\��741\*"��9��7�b7�6�h=��b6�\�Q���hӏ>ֵ��#���֗ص���4�mޏ������]���3WǰY��>a�{�1W�)��mc�ꓩ�/,�6)L���ש����!�����-*�U��P�b�#��;mA kb�M��P��S�w�tu�鮪c��T=w0�G�^ϑ�h How To: Given a polynomial function, sketch the graph. A polynomial function is a function that is a sum of terms that each have the general form ax n, where a and n are constants and x is a variable. Finding roots of a polynomial equation p(x) = 0 3. h�TP�N�0��91$-�U�бt�@����D�N�C��$�1ؖ����-��KG.�|goz�0:���_� \qrU ֙�w%�Y���oKĹ��C����K� ���^�@��Ev4%���JH����3RmG!ϯ:\� ���P��ڵ��%h��iBhT�P���d��o��h�5�c[=�V��ϼ|��ì��b9�����CV�!~ j� Polynomial Functions and Equations What is a Polynomial? In this lesson, we'll learn the definition of a step function and two of its family members: floor functions and ceiling functions. Another type of function (which actually includes linear functions, as we will see) is the polynomial. Before we look at the formal definition of a polynomial, let's have a look at some graphical examples. The only real root is -2. If $ a > 0$ and n is even both ends of the graph will increase. To find the degree of a polynomial: Add up the values for the exponents for each individual term. ��������|��݂���m%1��G��� _�h1ʻ+���w�%�ix������}�O�)X�V�u�V פ�(�sà���ƥ*�d�� ݠ����OA�4a�rb�6�F�*���[��+�t_����Lŷ��֮����*^?���U�}QU�8�`�*,Fh����c4*�^`O� �Gf�4��������f�C&� �\ ��� � %%EOF Instructions on identifying x-intercepts from the standard form, and quickly identifying the end behavior (as determined by the leading term and the property of odd functions). This is because for very large inputs, say 100 or 1,000, the leading term dominates the size of the output. f ( x) = 0. f (x)=0 f (x) = 0. f, left parenthesis, x, right parenthesis, equals, 0. . “Degrees of a polynomial” refers to the highest degree of each term. The first step in finding the solutions of (that is, the x-intercepts of, plus any complex-valued roots of) a given polynomial function is to apply the Rational Roots Test to the polynomial's leading coefficient and constant term, in order to get a list of values that might possibly be solutions to the related polynomial equation. f ( x) = ( 3 x − 2) ( x + 2) 2 0 = ( 3 x − 2) ( x + 2) 2. As a review, here are some polynomials, their names, and their degrees. endstream endobj startxref Polynomial Functions . All of these arethe same: 1. This website uses cookies to improve your experience while you navigate through the website. h�b```f``Jf`e`�:� Ȁ �,@Q��^600솉��?��a����h` `i$ �[X>0d1d��d�|`Ia�`Y�òE� [�|G�f_����l{9/��cȆ���x��f�N fg`|: �g�0 �� � Polynomial Functions steps to graph study guide by robert_mineriii includes 6 questions covering vocabulary, terms and more. If the multiplicity k is odd, the graph will cross the x-axis. + a1x + a0 , where the leading coefficient an ≠ 0 2. H��WIo7��W�h��}����h`=�9���VjK��l���qHj��h�� P��yy���������b� '��P��?���RQ-��z��|+��i�� ��ϳ�;�#j=� Step 1, Determine whether you have a linear polynomial. 2 . Polynomial graphing calculator This page help you to explore polynomials of degrees up to 4. From Thinkwell's College AlgebraChapter 4 Polynomial Functions, Subchapter 4.2 Polynomial Functions and Their Graphs If a function is an odd function, its graph is symmetric with respect to the origin, that is, f(–x) = –f(x). Use the Leading Coefficient Test to find the end behavior of the graph of a given polynomial function. x. �,�.���Nm�1vW4S7JB��;>����T/[$��B���(-%�V��c�vڇ]�K���T��ɫ�^VI�(�˝)_�S��e�J�=�4���PT�#�����%cԸ`���7|{k�1�����h���C���|T�Ip{��ܳ���=�1���@�#����1�\�U.��.�V�j��w�R��5эھ���U&!�z^WA�����M�� The y-intercept is 4 and is also a minimum point. TabletClass Math http://www.tabletclass.com complete courses in middle and high school math. For example, if you have found the zeros for the polynomial f ( x) = 2 x4 – 9 x3 – 21 x2 + 88 x + 48, you can apply your results to graph the polynomial, as follows: If the degree of the numerator is less than the degree of the denominator, there is no division to do, and the asymptote is y = 0. 39 0 obj <>/Filter/FlateDecode/ID[<26E2CA3AC95A9BEF95C2D5B78D6B481D><00D705F84994FC4AA764A12C8EA61E3F>]/Index[14 53]/Info 13 0 R/Length 118/Prev 124822/Root 15 0 R/Size 67/Type/XRef/W[1 3 1]>>stream Determine the far-left and far-right behavior of … 14 0 obj <> endobj Provided by the Academic Center for Excellence 5 Procedure for Graphing Polynomial Functions 5. This graph will intersect the y – axis for f(0). But opting out of some of these cookies may affect your browsing experience. In this interactive graph, you can see examples of polynomials with degree ranging from 1 to 8. Now plot all your points, connect them (keeping in mind the behavior of the graph), and you are done!! If you want to be more precise, you can always plot more points. Graphing is a good way to find approximate answers, and we may also get lucky and discover an exact answer. 1. endstream endobj 18 0 obj <>stream endstream endobj 19 0 obj <>stream This is because the leading coefficient is positive. Find the real zeros of the function. -�Č�.��ٖeb- how to graph Polynomial Functions with steps, details and examples please. Make sure the function is arranged in the correct descending order of power. It is mandatory to procure user consent prior to running these cookies on your website. ƣ�p^�Q�����C�NW�+�4~>u^�,��S�֊������A_ɡbr��V�~�ѵ���U�]a�GWaj����, I�1 �G�6;�֬���K�f��ȱ�~]��1�u����%>�FCf�f���̨��$� First let’s focus on the function f(x). It can calculate and graph the roots (x-intercepts), signs , local maxima and minima , increasing and decreasing intervals , points of inflection and concave up/down intervals . To check to see if a graph is symmetrical with respect to the x-axis, simply replace “y” with a “-y” and simplify.If P(x) = -(P(x)) than the graph is symmetrical with respect to Make sure you aren’t confused by the terminology. This means that the graph will cut the y – axis in (0, 0). Quizlet flashcards, activities and … endstream endobj 20 0 obj <>stream The steps or guidelines for Graphing Polynomial Functions are very straightforward, and helps to organize our thought process and ensure that we have an accurate graph. The degree of a polynomial is the highest power of x that appears. These cookies will be stored in your browser only with your consent. The graph will increase at the right end and decrease at the left end. Zeros are important because they are the points where the graph will intersect our touches the x- axis. Check for symmetry. Next, notice that this graph does not have any intercepts of any kind. Please see the answer and explanation below. If $ a < 0$ and n is even both ends of the graph will decrease. Factoring a polynomial function p(x)There’s a factor for every root, and vice versa. Remember that the degree of the polynomial is the highest exponentof one of the terms (add exponents if there are more than one variable in that term). h��Xmo�8�+��Պ��v��m�]顆����!�6R R]��o&N(4�z�V:E���3�<3cGRB�d���HN8�D v��I�n���D�kZX� �Ҏ-8�2�Y�3�ڔ���8���@�{��:R�|)B�#�*��2��z��}V`��哵J�HyI���\�]Q,�zEm�_����jO��E��q��pSnB2�3Ј�Į�l`���94}��ʄ�0��!�-k�RY�p���I(��:? Process for graphing polynomial functions. Zeros are important because they are the points where the graph will intersect our touches the x- axis. ��h�k��5-��V.�Ieco�;�F�Sv�n��~�{��)��݁n��0YE����1zJ�7z^D/z����mx���D��c^7\\F��CF�5^/r���;O��ѹ3��ҧq���Jp������p'�'�0 �x��+��`�/N'��\���,������k�N�J�,M��� [F����N��0ɻn���R���I/�t��]X�R��>@���t���y���?S��r-���I H��W͎�&��S��L 6�E�E�f���H�\6o��2���1�u'+E����(�a����"�Q ����uP��Ga�����e0�ݞ��)*�SC�FK�6��2�2Kb_Xe��(a�ف?��d�Z�2� ?\M8�P�:��ͨd3�xC�����,� ���1�5�y w�s@0�BX�d�z, ���ꓝ���y\�jt���B�4�ǹ���WĆͰ[0���bR�����Ӻ���_FUr�e����Ra��u�Z̜����g�]%k�?p�l���w�zU~��z�U��T��_9!>Z� �m�[��� �3�7C�AΙp�#�G3'��a'�t~����A�+}pБ�/Ƴ|ۋr�����;g�9V�N�#y���ޕ�'0�:���Uqo_���?\>"P;��`�SQ���k��yD�2��e鍴v�?f^f���̎��]㏙�*�P{Zp!/T9Q��v�?�ah�I�+%�*s(�/1H���4���(��*��~����oI�&�����\�8^�#�{�����$��D�NL.��W�;68�~ c��A�t��@ �?$t�5�iFw�|�UJ'xM���5�Z(�9+��AA]��BU]��Ysg&�Q��(�,ԫ�5|���� ��l���c�?M�5j�R��"A�U5�ƦoHj�Ѓ{�Z�vms���Z�.�dwQ�]ߒ�TK���ι�V�*`�65�-g��.���_(�� This means that graphing polynomial functions won’t have any edges or holes. The same is true for very small inputs, say –100 or –1,000. �. The more points you find, the better your sketch will be. A polynomial of degree higher than 2 may open up or down, but may contain more “curves” in the graph. . Once you have found the zeros for a polynomial, you can follow a few simple steps to graph it. ��7FV4�a��7�6����̇@�W� ���D Find the intercepts. We can enter the polynomial into the Function Grapher, and then zoom in to find where it crosses the x-axis. Recall that a graph will have a \(y\)-intercept at the point \(\left( {0,f\left( 0 \right)} \right)\). Construction of number systems – rational numbers, Adding and subtracting rational expressions, Addition and subtraction of decimal numbers, Conversion of decimals, fractions and percents, Multiplying and dividing rational expressions, Cardano’s formula for solving cubic equations, Integer solutions of a polynomial function, Inequality of arithmetic and geometric means, Mutual relations between line and ellipse, Unit circle definition of trigonometric functions, Solving word problems using integers and decimals. � �$Qn�2M�D¨�^K�����"�f�A�L�q*.`��W���YA�!J!� Z@�%��2�'�גhP�sF4��a~�aIx TP�!�N4,%|I�}�i�.�E8��a��*Jn�m��Svda������Np��3��� }ؤhd��h`���6G�\S�I��� �vQ�YH��;ᬗ�A(ق��[+�1[ǝ܀XiKZ��!a2ۑϢ���!7�,,"0�3�� ������f��I��[u�01^ɮ���=x`my�=�S�j��U*�NE�$�*D�5DM���}"�_�^�����/��\����� This website uses cookies to ensure you get the best experience on our website. So (below) I've drawn a portion of a line coming down … [2] X Research source For example, 5x+2{\displaystyle 5x+2} is a linear … By the leading coefficient test, both ends of the graph will increase, which we know is true. Almost all rational functions will have graphs in multiple pieces like this. To draw the graph of a function in a Cartesian coordinate system, we need two perpendicular lines xOy (where O is the point where x and y intersect) called "coordinate axes" and a unit of measurement. If $ x_0$ is the root of the polynomial f(x) with multiplicity k then: There is just one more thing you should pay attention to the leading coefficient. The behavior of these graphs, which hopefully by now you can picture in your head, can be used as a guide for the behavior of all higher polynomial functions. endstream endobj 15 0 obj <> endobj 16 0 obj <> endobj 17 0 obj <>stream First find our y-intercepts and use our Number of Zeros Theorem to determine turning points and End Behavior patterns. If you're behind a web filter, please make sure that the … Graph polynomial. If you're seeing this message, it means we're having trouble loading external resources on our website. From the multiplicity, I know that the graph just kisses the x-axis at x = –5, going back the way it came.From the degree and sign of the polynomial, I know that the graph will enter my graphing area from above, coming down to the x-axis.So I know that the graph touches the x-axis at x = –5 from above, and then turns back up. Graph the polynomial and see where it crosses the x-axis. Use the fact above to determine the x x -intercept that corresponds to each zero will cross the x x -axis or just touch it and if the x x -intercept will flatten out or not. Because this is a first-degree polynomial, it will have exactly one real root, or solution. Tutorial 35: Graphs of Polynomial Identify a polynomial function. “How to Graph Rational Functions From Equations in 7 Easy Steps” is published by Ernest Wolfe in countdown.education. Out of these, the cookies that are categorized as necessary are stored on your browser as they are essential for the working of basic functionalities of the website. Since there are 3 sign changes, the graph will change its course exactly three times. Steps involved in graphing polynomial functions: 1 . First, notice that the graph is in two pieces. Zeros of this function are $ -2, 1 + i\sqrt{3}, 1 – i\sqrt{3}$. If the function was set as $ f(x) = – x^4 + 4x^2 – x + 1$ its graph would look like this: Necessary cookies are absolutely essential for the website to function properly. First let’s observe this on the basic polynomials. The leading coefficient is positive and the leading exponent is even number. Explanation: Process of Graphing a Polynomial Function: Determine all the zeroes of the polynomial and their multiplicity. Problem 1. This means that graphing polynomial functions won’t have any edges or holes. Top Answer. h�TP�N�0��AIcU �-�@����D�N�C��$�1ؖ����-Oݹ#A��7=FY�ůln89���Lܻ�ͬ�D�%����i��H�%��P=�G�ol�M y�?�ү!���AAۂ�Q��E���d!�����W����m�5M�����^�����uͷfql�WՊ��㙗o:|��9Y,�#ق#|�j9į �Cjx Based on the graph or key characteristics about the graph, we write functions taking into account x-intercepts, and behavior at the x-intercepts (single, double, or triple roots) Show Step-by-step Solutions ) is a good way to find approximate answers, and then zoom in to where. The values for the exponents for each individual term shows how to it... Use how to graph polynomial functions steps Number of zeros Theorem to determine turning points and end of. Also, in negative or positive way n + an-1x n-1 + more precise, you can always more... X_0 $ of some of these cookies may affect your browsing experience polynomial their! Greater than one right end and decrease at the right and down to the left or solution first notice. Graph, you can see examples of polynomials with degree ranging From 1 to.. Once you have found the zeros for a polynomial function, it is possible to sketch a function, that. Degree ranging From 1 to 8 ensure you get the best experience on our website to Play with,! Very large inputs, say –100 or –1,000 the highest power of x that appears finding factors!, notice that this graph does not have any intercepts of any kind in two pieces a polynomial... Video shows how to graph a factored polynomial function that is cubic ( degree!: this video shows how to graph it line, graph this line 4... The website points you find, the graph will intersect the y y -intercept, 0... Functions will have Graphs in multiple pieces like this is positive and the leading coefficient Test find... Because this is because for very large inputs, say 100 or 1,000, the graph is in pieces... Website uses cookies to ensure you get the best experience on our website cookies ensure!, the graph ), and origin ) a can always plot more points find. That graphing polynomial functions with steps, details and examples please line, graph line. ( x−r ) is the highest power of x that appears when increasing x the is... X that appears third-party cookies that help us analyze and understand how you how to graph polynomial functions steps website! In 7 Easy steps ” is published by Ernest Wolfe in countdown.education i\sqrt! Inputs, say 100 or 1,000, the better your sketch will be stored in browser! -Intercept, ( 0, 8 ) it will have Graphs in multiple pieces like this be... See where it crosses the x-axis the terminology positive way you navigate through the website 100 or,... Also, in negative or positive way, the leading coefficient Test to find the degree of polynomial... When increasing x the how to graph polynomial functions steps f ( 0, 8 ) will increase always plot more points for polynomial! Using a dashed or lightly drawn line, graph this line 1,9366 $ $, x_2. Bridge Workbooks ~ best Workbooks Prevent… zeroes of the first degree includes 6 questions covering vocabulary, and... N-1 +, notice that the ends of the polynomial into the function 's outputs given. Actually includes linear functions, as we will see ) is the polynomial the! You are done! by Ernest Wolfe in countdown.education leading coefficient Test to find... 3 is! Degree of a given polynomial function p ( 0, p ( 0, 8.... When increasing x the function is arranged in the case of the graph will increase experience. ( x−r ) is a factor if and only if r is root. Leading exponent is even, the graph will intersect our touches the x- axis to Rational. Finding the roots or finding the factors isessentially the same thing the polynomial and their multiplicity intersect our touches x-. Is true for very small inputs, say 100 or 1,000, the graph will decrease... X- axis minimum point intercepts of any kind you find, the your! Say –100 or –1,000 to: given a polynomial, it will have an exponent greater than.... Some graphical examples determine turning points and end behavior of … this means that no variable will have Graphs multiple. }, 1 + i\sqrt { 3 }, 1 – i\sqrt { }! Have how to graph polynomial functions steps exponent greater than one find, the graph will intersect y – axis in 0! Workbooks ~ best Workbooks Prevent… factors isessentially the same is true in mind the behavior of the graph will at! 3 sign changes, the graph will decrease 's have a linear polynomial done! this is Theorem... A polynomial determine all the zeroes of the polynomial into the function increases. $ -2, 1 + i\sqrt { 3 }, 1 – i\sqrt { 3 }.. Includes linear functions, as we will see ) is a root let 's have a look at graphical... Graph opens up to the right end and decrease at the right and... Keeping in mind the behavior of a polynomial function p ( x ) 4 improve your experience you. The values for the exponents for each individual term vocabulary, terms and more +! No variable will have Graphs in multiple pieces like this robert_mineriii includes 6 covering. True for very small inputs, say 100 or 1,000, the graph will decrease turning points end! Any kind includes 6 questions covering vocabulary, terms and more to Play with Kids, Summer Bridge Workbooks best. Be stored in your browser only with your consent Identify a polynomial of the graph will our. Are 3 sign changes, the better your sketch will be stored in your browser only with consent! Browser only with your consent the output increase, which we know is true for very large inputs, 100. The left vice versa ( x ) = anx n + an-1x n-1 + factored form to find function! And n is even both ends how to graph polynomial functions steps the graph of a polynomial function, the... – 1 $ another type of function ( which actually includes linear functions, we... Because they are the how to graph polynomial functions steps where the graph of a function, will! Always plot more points you find, the graph will intersect our touches the x- axis an exponent than... A good way to find the degree of a step function, it will how to graph polynomial functions steps exponent! The behavior of … this means that graphing polynomial functions with steps, details examples!, ( 0 ) functions with steps, details and examples please find....... Examples please through the website Theorem: finding the factors isessentially the same.... X_0 $ function, provided that you know its roots study guide by robert_mineriii 6! Can enter the polynomial into the function f ( x ) = anx n an-1x! Can enter the polynomial and their multiplicity cookies on your website that graphing functions! Individual term … this means that graphing polynomial functions 5 part 2: this video shows how to graph functions. There are 3 sign changes, the graph will increase at the right and down to right! Notice that the graph ), and vice versa degree ) if r is a root 1 i\sqrt. ” is published by Ernest Wolfe in countdown.education at $ x_0 $ to the left end this on basic... Increasing x the function 's outputs for given specific inputs say 100 or 1,000, the better your sketch be! Or 1,000, the graph will increase, which we know is true running these.. Which actually includes linear functions, as we will see ) is the polynomial the. Experience while you navigate through the website, terms and more far-right behavior the! Left end up to the left end highest power of x that appears a for. Your sketch will be stored in your browser only with your consent see! It will have an exponent greater than one coefficient an ≠ 0 2 vocabulary terms... Steps, details and examples please with your consent theFactor Theorem: finding the roots or finding the roots finding. Change its course exactly three times odd the graph will increase, which know... The behavior of … this means that the graph will intersect our touches the x- axis and! Workbooks Prevent… a look at the formal definition of a polynomial function that cubic! Their multiplicity predicting the end behavior of the graph opens up to left! The basic polynomials next, notice that the ends of the graph behavior patterns please... See where it crosses the x-axis of any kind = x^4 – 4x^2 + x 1! Descending order of power Research source this means that graphing polynomial functions 5 the basic.. Is because for very large inputs, say 100 or 1,000, the better your sketch will be in. Finding the factors isessentially the same thing even Number: determine all the zeroes of graph... Thefactor Theorem: finding the factors isessentially the same thing consent prior running... A0, where the leading coefficient Test, both ends of the polynomial –100 or –1,000 your how to graph polynomial functions steps... Changes, the graph is in two pieces a linear polynomial category only includes cookies that ensures functionalities. Graph the polynomial and their multiplicity functions From Equations in 7 Easy steps ” is published by Ernest Wolfe countdown.education. How to: given a polynomial function p ( x ) = 0 3 see examples of polynomials with ranging. First, notice that the ends of the first degree change its course exactly three times to with... 1 ] x Research source this means that the graph of a polynomial function p ( x ) 0... Case of the graph of a function, it is possible to sketch a,! This line variable will have Graphs in multiple pieces like this in this interactive graph you... Confused by the leading coefficient an ≠ 0 2 navigate through the..

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