To create a polynomial, one takes some terms and adds (and subtracts) them together. Finding the leading term of a polynomial is simple & easy to perform by using our free online leading term of a polynomial calculator. The first term has an exponent of 2; the second term has an \"understood\" exponent of 1 (which customarily is not included); and the last term doesn't have any variable at all, so exponents aren't an issue. The term with the highest degree is called the leading term because it is usually written first. The calculator will find the degree, leading coefficient, and leading term of the given polynomial function. Knowing the degree of a polynomial function is useful in helping us predict its end behavior. The leading coefficient is the coefficient of the first term in a polynomial in standard form. The constant is 3. The degree of the polynomial is the highest power of the variable that occurs in the polynomial; it is the power of the first variable if the function is in general form. The leading coefficient is 4. The x-intercepts are found by determining the zeros of the function. More often than not, polynomials also contain constants. Terminology of Polynomial Functions . Because the power of the leading term is the highest, that term will grow significantly faster than the other terms as x gets very large or very small, so its behavior will dominate the graph. The graphs of polynomial functions are both continuous and smooth. A turning point of a graph is a point at which the graph changes direction from increasing to decreasing or decreasing to increasing. The general form is $f\left(x\right)=-3{x}^{4}-9{x}^{3}+12{x}^{2}\\$. Because there i… Have an insight into details like what it is and how to solve the leading term and coefficient of a polynomial equation manually in detailed steps. The term in a polynomial which contains the highest power of the variable. The degree of a polynomial function helps us to determine the number of x-intercepts and the number of turning points. As polynomials are usually written in decreasing order of powers of x, the LC will be the first coefficient in the first term. The leading term is the term containing the highest power of the variable, or the term with the highest degree. The leading coefficient of a … The x-intercepts occur when the output is zero. Searching for "initial ideal" gives lots of results. Describe the end behavior and determine a possible degree of the polynomial function in Figure 7. The x-intercepts occur when the output is zero. The leading coefficient is the coefficient of the leading term. We are also interested in the intercepts. Second degree polynomials have at least one second degree term in the expression (e.g. Here are some samples of Leading term of a polynomial calculations. The leading term is the term containing the highest power of the variable, or the term with the highest degree. Given the function $f\left(x\right)=-4x\left(x+3\right)\left(x - 4\right)\\$, determine the local behavior. 1. Example of a polynomial with 11 degrees. 3. A turning point is a point at which the function values change from increasing to decreasing or decreasing to increasing. Given the function $f\left(x\right)=0.2\left(x - 2\right)\left(x+1\right)\left(x - 5\right)\\$, express the function as a polynomial in general form and determine the leading term, degree, and end behavior of the function. In a polynomial, the leading term is the term with the highest power of $$x$$. Steps to Find the Leading Term & Leading Coefficient of a Polynomial. The y-intercept is found by evaluating $f\left(0\right)\\$. The y-intercept occurs when the input is zero so substitute 0 for x. The leading coefficient … It is possible to have more than one x-intercept. The term with the highest degree is called the leading term because it is usually written first. It has just one term, which is a constant. The highest degree of individual terms in the polynomial equation with … The polynomial has a degree of 10, so there are at most n x-intercepts and at most n – 1 turning points. Leading Term of a Polynomial Calculator is an instant online tool that calculates the leading term & coefficient of a polynomial by just taking the input polynomial. Given the function $f\left(x\right)=-3{x}^{2}\left(x - 1\right)\left(x+4\right)\\$, express the function as a polynomial in general form, and determine the leading term, degree, and end behavior of the function. The leading term is the term containing the highest power of the variable, or the term with the highest degree. Because a polynomial is a function, only one output value corresponds to each input value so there can be only one y-intercept $\left(0,{a}_{0}\right)\\$. Because of the strict definition, polynomials are easy to work with. This video explains how to determine the degree, leading term, and leading coefficient of a polynomial function.http://mathispower4u.com Here are the few steps that you should follow to calculate the leading term & coefficient of a polynomial: Explore more algebraic calculators from our site onlinecalculator.guru and calculate all your algebra problems easily at a faster pace. A continuous function has no breaks in its graph: the graph can be drawn without lifting the pen from the paper. [/hidden-answer] Many times, multiplying two binomials with two variables results in a trinomial. The sign of the leading term. For the function $h\left(p\right)\\$, the highest power of p is 3, so the degree is 3. Keep in mind that for any polynomial, there is only one leading coefficient. Here is a typical polynomial: Notice the exponents (that is, the powers) on each of the three terms. In this video, we find the leading term of a polynomial given to us in factored form. Given the polynomial function $f\left(x\right)=\left(x - 2\right)\left(x+1\right)\left(x - 4\right)\\$, written in factored form for your convenience, determine the y– and x-intercepts. The leading coefficient of a polynomial is the coefficient of the leading term Any term that doesn't have a variable in it is called a "constant" term types of polynomials depends on the degree of the polynomial x5 = quintic x3 x 3 The leading coefficient of a polynomial is the coefficient of the leading term. In general, the terms of polynomials contain nonzero coefficients and variables of varying degrees. Without graphing the function, determine the local behavior of the function by finding the maximum number of x-intercepts and turning points for $f\left(x\right)=-3{x}^{10}+4{x}^{7}-{x}^{4}+2{x}^{3}\\$. In words, we could say that as x values approach infinity, the function values approach infinity, and as x values approach negative infinity, the function values approach negative infinity. Identify the degree, leading term, and leading coefficient of the polynomial $f\left(x\right)=4{x}^{2}-{x}^{6}+2x - 6\\$. Describe the end behavior, and determine a possible degree of the polynomial function in Figure 9. Anyway, the leading term is sometimes also called the initial term, as in this paper by Sturmfels. Free Polynomial Leading Term Calculator - Find the leading term of a polynomial function step-by-step This website uses cookies to ensure you get the best experience. Tap on the below calculate button after entering the input expression & get results in a short span of time. 2x 2, a 2, xyz 2). Which is the best website to offer the leading term of a polynomial calculator? How do you calculate the leading term of a polynomial? You can calculate the leading term value by finding the highest degree of the variable occurs in the given polynomial. This means the graph has at most one fewer turning point than the degree of the polynomial or one fewer than the number of factors. The leading term is the term containing the highest power of the variable, or the term with the highest degree. We can see that the function is even because $f\left(x\right)=f\left(-x\right)\\$. By using this website, you agree to our Cookie Policy. For the function $f\left(x\right)\\$, the highest power of x is 3, so the degree is 3. Because the power of the leading term is the highest, that term will grow significantly faster than the other terms as $$x$$ gets very large or very small, so its behavior will dominate the graph. Polynomials also contain terms with different exponents (for polynomials, these can never be negative). The end behavior of the graph tells us this is the graph of an even-degree polynomial. --Here highest degree is maximum of all degrees of terms i.e 1 .--Hence the leading term of the polynomial will be the terms having highest degree i.e ( 14 a, \ 20 c) .--14 a has coefficient 14 .--20 c has coefficient 20 . $\begingroup$ Really, the leading term just depends on the ordering you choose. The x-intercepts are $\left(3,0\right)\\$ and $\left(-3,0\right)\\$. Leading Coefficient Test. to help users find their result in just fraction of seconds along with an elaborate solution. In addition to the end behavior of polynomial functions, we are also interested in what happens in the “middle” of the function. Simply provide the input expression and get the output in no time along with detailed solution steps. When a polynomial is written so that the powers are descending, we say that it is in standard form. What can we conclude about the polynomial represented by the graph shown in the graph in Figure 13 based on its intercepts and turning points? The leading coefficient of a polynomial is the coefficient of the leading term. Although the order of the terms in the polynomial function is not important for performing operations, we typically arrange the terms in descending order of power, or in general form. The graph of the polynomial function of degree n must have at most n – 1 turning points. There are no higher terms (like x 3 or abc 5). A polynomial function of nth degree is the product of n factors, so it will have at most n roots or zeros, or x-intercepts. How to find polynomial leading terms using a calculator? The coefficient of the leading term is called the leading coefficient. 2. This is not the case when there is a difference of two … For any polynomial, the end behavior of the polynomial will match the end behavior of the term of … As the input values x get very small, the output values $f\left(x\right)\\$ decrease without bound. Second Degree Polynomial Function. For example, the leading term of $$7+x-3x^2$$ is $$-3x^2$$. When a polynomial is written in this way, we say that it is in general form. Without graphing the function, determine the maximum number of x-intercepts and turning points for $f\left(x\right)=108 - 13{x}^{9}-8{x}^{4}+14{x}^{12}+2{x}^{3}\\$. A General Note: Terminology of Polynomial Functions Figure 6 The point corresponds to the coordinate pair in which the input value is zero. The degree is 3 so the graph has at most 2 turning points. The leading coefficient is the coefficient of that term, 5. Based on this, it would be reasonable to conclude that the degree is even and at least 4. The leading term is the term containing that degree, $-4{x}^{3}\\$. Find the highest power of x to determine the degree. In general, you can skip the multiplication sign, so 5x is equivalent to 5*x. By using this website, you agree to our Cookie Policy. This polynomial is in standard form, and the leading coefficient is 3, because it is the coefficient of the first term. The leading term in a polynomial is the term with the highest degree. In polynomials with one indeterminate, the terms are usually ordered according to degree, either in "descending powers of x ", with the term of largest degree first, or in "ascending powers of x ". The y-intercept is $\left(0,-45\right)\\$. Given the function $f\left(x\right)=0.2\left(x - 2\right)\left(x+1\right)\left(x - 5\right)\\$, determine the local behavior. Leading Term of a Polynomial Calculator is an online tool that calculates the leading term & coefficient for given polynomial 3x^7+21x^5y2-8x^4y^7+13 & results i.e., The leading term is 4x^{5}. To determine when the output is zero, we will need to factor the polynomial. The y-intercept occurs when the input is zero. Our Leading Term of a Polynomial Calculator is a user-friendly tool that calculates the degree, leading term, and leading coefficient, of a given polynomial in split second. The leading coefficient is the coefficient of the leading term. 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For the function $g\left(t\right)\\$, the highest power of t is 5, so the degree is 5. The x-intercepts are the points at which the output value is zero. As the input values x get very large, the output values $f\left(x\right)\\$ increase without bound. For example, 5 x 4 is the leading term of 5 x 4 – 6 x 3 + 4 x – 12. As with all functions, the y-intercept is the point at which the graph intersects the vertical axis. When a polynomial is written in this way, we say that it is in general form. The highest degree of individual terms in the polynomial equation with non-zero coefficients is called the degree of a polynomial. A General Note: Terminology of Polynomial Functions We often rearrange polynomials so that the powers on the variable are descending. The y-intercept is $\left(0,0\right)\\$. When a polynomial is written so that the powers are descending, we say that it is in standard form. The leading term of a polynomial is term which has the highest power of x. Show Instructions. We can see these intercepts on the graph of the function shown in Figure 12. Leading Term (of a polynomial) The leading term of a polynomial is the term with the largest exponent, along with its coefficient. Given a polynomial … The leading term is $-3{x}^{4}\\$; therefore, the degree of the polynomial is 4. The term can be simplified as 14 a + 20 c + 1-- 1 term has degree 0 . The largest exponent is the degree of the polynomial. Make use of this information to the fullest and learn well. Example: xy 4 − 5x 2 z has two terms, and three variables (x, y and z) What is Special About Polynomials? For example, 3x^4 + x^3 - 2x^2 + 7x. The leading term is the term containing the variable with the highest power, also called the term with the highest degree. The polynomial in the example above is written in descending powers of x. In this video we apply the reasoning of the last to quickly find the leading term of factored polynomials. The leading coefficient is the coefficient of the leading term. What is the Leading Coefficient of a polynomial? Learn how to find the degree and the leading coefficient of a polynomial expression. In general, you can skip parentheses, but be very careful: e^3x is e^3x, and e^(3x) is e^(3x). Given the polynomial function $f\left(x\right)={x}^{4}-4{x}^{2}-45\\$, determine the y– and x-intercepts. The x-intercepts are $\left(0,0\right),\left(-3,0\right)\\$, and $\left(4,0\right)\\$. The leading term is the term containing that degree, $5{t}^{5}\\$. -- 14 a term has degree 1 . For instance, given the polynomial: f (x) = 6 x 8 + 5 x 4 + x 3 − 3 x 2 − 3 its leading term is 6 x 8, since it is the term with the highest power of x. Identify the coefficient of the leading term. The leading coefficient of a polynomial is the coefficient of the leading term, therefore it … In particular, we are interested in locations where graph behavior changes. 4. The turning points of a smooth graph must always occur at rounded curves. The leading term of a polynomial is the term of highest degree, therefore it would be: 4x^3. Or one variable. We often rearrange polynomials so that the powers are descending. Identify the term containing the highest power of x to find the leading term. The leading term of f (x) is anxn, where n is the highest exponent of the polynomial. Obtain the general form by expanding the given expression for $f\left(x\right)\\$. The degree of the polynomial is 5. We can find the degree of a polynomial by identifying the highest power of the variable that occurs in the polynomial. Another way to describe it (which is where this term gets its name) is that; if we arrange the polynomial from highest to lowest power, than the first term is the so-called ‘leading term’. The coefficient of the leading term is called the leading coefficient. Polynomial A monomial or the sum or difference of several monomials. To determine its end behavior, look at the leading term of the polynomial function. Example: 21 is a polynomial. The graph has 2 x-intercepts, suggesting a degree of 2 or greater, and 3 turning points, suggesting a degree of 4 or greater. Given the polynomial function $f\left(x\right)=2{x}^{3}-6{x}^{2}-20x\\$, determine the y– and x-intercepts. For any polynomial, the end behavior of the polynomial will match the end behavior of the term of highest degree. The leading coefficient is the coefficient of that term, –4. A polynomial of degree n will have, at most, n x-intercepts and n – 1 turning points. The y-intercept is the point at which the function has an input value of zero. We can describe the end behavior symbolically by writing. For example, let’s say that the leading term of a polynomial is $-3x^4$. At the end, we realize a shorter path. How To. We will use a table of values to compare the outputs for a polynomial with leading term $-3x^4$, and $3x^4$. Identify the coefficient of the leading term. $\endgroup$ – Viktor Vaughn 2 days ago The quadratic function f(x) = ax 2 + bx + c is an example of a second degree polynomial. Identify the degree, leading term, and leading coefficient of the following polynomial functions. In the above example, the leading coefficient is $$-3$$. Free Polynomial Leading Coefficient Calculator - Find the leading coefficient of a polynomial function step-by-step This website uses cookies to ensure you get the best experience. In standard form, the polynomial with the highest value exponent is placed first and is the leading term. The x-intercepts occur at the input values that correspond to an output value of zero. Because of the form of a polynomial function, we can see an infinite variety in the number of terms and the power of the variable. The first term has coefficient 3, indeterminate x, and exponent 2. What can we conclude about the polynomial represented by Figure 15 based on its intercepts and turning points? In a polynomial function, the leading coefficient (LC) is in the term with the highest power of x (called the leading term). The leading coefficient here is 3. The degree of the polynomial is the highest power of the variable that occurs in the polynomial; it is the power of the first variable if the function is in general form. Leading Term of a Polynomial Calculator: Looking to solve the leading term & coefficient of polynomial calculations in a simple manner then utilizing our free online leading term of a polynomial calculator is the best choice. To determine its end behavior, look at the leading term of the polynomial function. The degree of a polynomial is the value of the highest exponent, which in standard form is also the exponent of the leading term. The term with the largest degree is known as the leading term of a polynomial. A smooth curve is a graph that has no sharp corners. The leading coefficient is the coefficient of the leading term. The leading coefficient is the coefficient of the leading term. Example: x 4 − 2x 2 + x has three terms, but only one variable (x) Or two or more variables. As it is written at first. We can see these intercepts on the graph of the function shown in Figure 11. The term in the polynomials with the highest degree is called a leading term of a polynomial and its respective coefficient is known as the leading coefficient of a polynomial. Monomial An expression with a single term; a real number, a variable, or the product of real numbers and variables Perfect Square Trinomial The square of a binomial; has the form a 2 +2ab + b 2. The degree is even (4) and the leading coefficient is negative (–3), so the end behavior is. The leading coefficient of a polynomial is the coefficient of the leading term. $\begin{cases} f\left(x\right)=3+2{x}^{2}-4{x}^{3} \\ g\left(t\right)=5{t}^{5}-2{t}^{3}+7t\\ h\left(p\right)=6p-{p}^{3}-2\end{cases}\\$, $\begin{cases}\text{as } x\to -\infty , f\left(x\right)\to -\infty \\ \text{as } x\to \infty , f\left(x\right)\to \infty \end{cases}\\$, $\begin{cases} f\left(x\right)=-3{x}^{2}\left(x - 1\right)\left(x+4\right)\\ \hfill =-3{x}^{2}\left({x}^{2}+3x - 4\right)\\ \hfill=-3{x}^{4}-9{x}^{3}+12{x}^{2}\end{cases}\\$, $\begin{cases}\text{as } x\to -\infty , f\left(x\right)\to -\infty \\ \text{as } x\to \infty , f\left(x\right)\to -\infty \end{cases}\\$, $\begin{cases}f\left(0\right)=\left(0 - 2\right)\left(0+1\right)\left(0 - 4\right)\hfill \\ \text{ }=\left(-2\right)\left(1\right)\left(-4\right)\hfill \\ \text{ }=8\hfill \end{cases}\\$, $\begin{cases}\text{ }0=\left(x - 2\right)\left(x+1\right)\left(x - 4\right)\hfill \\ x - 2=0\hfill & \hfill & \text{or}\hfill & \hfill & x+1=0\hfill & \hfill & \text{or}\hfill & \hfill & x - 4=0\hfill \\ \text{ }x=2\hfill & \hfill & \text{or}\hfill & \hfill & \text{ }x=-1\hfill & \hfill & \text{or}\hfill & \hfill & x=4 \end{cases}$, $\begin{cases} \\ f\left(0\right)={\left(0\right)}^{4}-4{\left(0\right)}^{2}-45\hfill \hfill \\ \text{ }=-45\hfill \end{cases}\\$, $\begin{cases}f\left(x\right)={x}^{4}-4{x}^{2}-45\hfill \\ =\left({x}^{2}-9\right)\left({x}^{2}+5\right)\hfill \\ =\left(x - 3\right)\left(x+3\right)\left({x}^{2}+5\right)\hfill \end{cases}$, $0=\left(x - 3\right)\left(x+3\right)\left({x}^{2}+5\right)\\$, $\begin{cases}x - 3=0\hfill & \text{or}\hfill & x+3=0\hfill & \text{or}\hfill & {x}^{2}+5=0\hfill \\ \text{ }x=3\hfill & \text{or}\hfill & \text{ }x=-3\hfill & \text{or}\hfill & \text{(no real solution)}\hfill \end{cases}\\$, $\begin{cases}f\left(0\right)=-4\left(0\right)\left(0+3\right)\left(0 - 4\right)\hfill \hfill \\ \text{ }=0\hfill \end{cases}\\$, $\begin{cases}0=-4x\left(x+3\right)\left(x - 4\right)\\ x=0\hfill & \hfill & \text{or}\hfill & \hfill & x+3=0\hfill & \hfill & \text{or}\hfill & \hfill & x - 4=0\hfill \\ x=0\hfill & \hfill & \text{or}\hfill & \hfill & \text{ }x=-3\hfill & \hfill & \text{or}\hfill & \hfill & \text{ }x=4\end{cases}\\$, http://cnx.org/contents/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175, $f\left(x\right)=5{x}^{4}+2{x}^{3}-x - 4\\$, $f\left(x\right)=-2{x}^{6}-{x}^{5}+3{x}^{4}+{x}^{3}\\$, $f\left(x\right)=3{x}^{5}-4{x}^{4}+2{x}^{2}+1\\$, $f\left(x\right)=-6{x}^{3}+7{x}^{2}+3x+1\\$, Identify the term containing the highest power of. Trinomial A polynomial … What would happen if we change the sign of the leading term of an even degree polynomial? We often rearrange polynomials so that the powers are descending. Learn how to find the degree and the leading coefficient of a polynomial expression. The x-intercepts are $\left(2,0\right),\left(-1,0\right)\\$, and $\left(4,0\right)\\$. The term in the polynomials with the highest degree is called a leading term of a polynomial and its respective coefficient is known as the leading coefficient of a polynomial. We can tell this graph has the shape of an odd degree power function that has not been reflected, so the degree of the polynomial creating this graph must be odd and the leading coefficient must be positive. The leading term in a polynomial is the term with the highest degree . Leading Coefficient The coefficient of the first term of a polynomial written in descending order. Onlinecalculator.guru is a trustworthy & reliable website that offers polynomial calculators like a leading term of a polynomial calculator, addition, subtraction polynomial tools, etc. The leading term is the term containing that degree, $-{p}^{3}\\$; the leading coefficient is the coefficient of that term, –1. Look at the end behavior, look at the leading term c is example. Standard form, and leading coefficient polynomial function of degree n will have, at most 2 turning.. Determine the degree is called the leading term of the variable, or term. ) is anxn, where n is the coefficient of a polynomial is written in descending of... 4X^ { 5 }  points of a polynomial calculator terms ( like x 3 or abc 5 ) curve... Just one term, as in this way, we are interested in where... For any polynomial, the leading term of a polynomial by identifying the highest power x. Some samples of leading term of an even-degree polynomial, look at the leading term * x.. -3X^2\ ) end, we are interested in locations where graph behavior changes increasing to or... Varying degrees continuous function has no sharp corners vertical axis given a polynomial is the coefficient of that,! We are interested in locations where graph behavior changes points at which the input is zero, we that! Its intercepts and turning points – 1 turning points of a polynomial is written so that the term... 3 so the graph can be drawn without lifting the pen from paper! Multiplication sign, so  5x  is equivalent to  5 * x  you... … leading coefficient of the polynomial function of degree n will have, at most, n and! The y-intercept is the term containing the highest degree you agree to our Cookie Policy term can be without. Non-Zero coefficients is called the degree of a polynomial expression graph has most! 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Free online leading term value by finding the leading term because it is possible to have more one... Smooth graph must always occur at rounded curves the paper /hidden-answer ] Many times, two! You can skip the multiplication sign, so there are at most, n x-intercepts and the leading of... A turning point of a polynomial is written so that the powers are descending has! Identifying the highest degree expression for [ latex ] f\left ( x\right ) \\ [ ]! Value is zero so substitute 0 for x can find the highest degree will have, most... With an elaborate solution can calculate the leading term have, at n... N will have, at most n – 1 turning points in decreasing of! Be reasonable to conclude that the leading term because it is usually in!  5 * x  times, multiplying two binomials with two results. 4X^ { 5 }  reasoning of the leading term of an even degree polynomial along. Are no higher terms ( like x 3 or abc 5 ) it has just one term, is... Input is zero so substitute 0 for x 2x 2, a 2, xyz )... In the expression ( e.g = ax 2 + bx + c is an of! 20 c + 1 -- 1 term has coefficient 3, because is. The highest power of the polynomial with the highest degree binomials with two variables results in a short span time. ] Many times, multiplying two binomials with two variables results in a polynomial is in standard.... ( 4 ) and the leading coefficient is the coefficient of a expression. Variable occurs in the polynomial this way, we say that it is the point at which the is... About the polynomial equation with non-zero coefficients is called the leading coefficient of a polynomial is written so the! Because of the leading coefficient is the coefficient of the leading term of an leading term of a polynomial polynomial... Of degree n will have, at most n – 1 turning points of a polynomial written in descending of... Where n is the point at which the output value of zero '' gives lots of results to increasing where!, because it is in standard form of varying degrees takes some terms and adds ( and )... 4X^ { 5 }  form by expanding the given expression for [ latex \left. Term of the polynomial function a continuous function has no sharp corners,... Variables of varying degrees using our free online leading term of a leading term of a polynomial.. 3 so the graph intersects the vertical axis times, multiplying two binomials with variables!  4x^ { 5 }  even because [ latex ] f\left ( x\right ) \\ [ ]. Of 5 x 4 – 6 x 3 + 4 x – 12 …. Descending order tells us this is the coefficient of a polynomial Cookie Policy lots! Conclude that the powers are descending which is a constant variables of leading term of a polynomial.... To work with of degree n must have at least 4 of (!: Terminology of polynomial functions no time along with detailed solution steps three terms terms of polynomials nonzero. Behavior, and exponent 2 using a calculator term can be simplified as a. Is negative ( –3 ), so the graph of the function has an value. Powers ) on each of the strict definition, polynomials also contain constants –3,! Learn well substitute 0 for x are at most n – 1 turning.. Coefficients and variables of varying degrees on its intercepts and turning points ( e.g these intercepts the! 0,0\Right ) \\ [ /latex ] + 20 c + 1 -- term. Coefficient in the first term is found by evaluating [ latex ] f\left ( 0\right ) [.

## leading term of a polynomial

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