Find the equation of the degree 4 polynomial f graphed below. This is particularly useful if you are new to fourth-degree equations or need to refresh your math knowledge as the 4th degree equation calculator will accurately compute the calculation so you can check your own manual math calculations. The factors of 1 are [latex]\pm 1[/latex]and the factors of 4 are [latex]\pm 1,\pm 2[/latex], and [latex]\pm 4[/latex]. The zeros of a polynomial calculator can find all zeros or solution of the polynomial equation P (x) = 0 by setting each factor to 0 and solving for x.
Polynomials: Sums and Products of Roots - mathsisfun.com The other zero will have a multiplicity of 2 because the factor is squared. Fourth Degree Polynomial Equations | Quartic Equation Formula ax 4 + bx 3 + cx 2 + dx + e = 0 4th degree polynomials are also known as quartic polynomials.It is also called as Biquadratic Equation. The equation of the fourth degree polynomial is : y ( x) = 3 + ( y 5 + 3) ( x + 10) ( x + 5) ( x 1) ( x 5.5) ( x 5 + 10) ( x 5 + 5) ( x 5 1) ( x 5 5.5) The figure below shows the five cases : On each one, they are five points exactly on the curve and of course four remaining points far from the curve. Use the Fundamental Theorem of Algebra to find complex zeros of a polynomial function. It's an amazing app! In other words, if a polynomial function fwith real coefficients has a complex zero [latex]a+bi[/latex],then the complex conjugate [latex]a-bi[/latex]must also be a zero of [latex]f\left(x\right)[/latex]. It can be written as: f (x) = a 4 x 4 + a 3 x 3 + a 2 x 2 +a 1 x + a 0. Find more Mathematics widgets in Wolfram|Alpha. of.the.function). Here is the online 4th degree equation solver for you to find the roots of the fourth-degree equations. Quartic Polynomials Division Calculator.
Quartic equation Calculator - High accuracy calculation We can use the Division Algorithm to write the polynomial as the product of the divisor and the quotient: [latex]\left(x+2\right)\left({x}^{2}-8x+15\right)[/latex], We can factor the quadratic factor to write the polynomial as, [latex]\left(x+2\right)\left(x - 3\right)\left(x - 5\right)[/latex]. Please enter one to five zeros separated by space. [latex]\begin{array}{l}\\ 2\overline{)\begin{array}{lllllllll}6\hfill & -1\hfill & -15\hfill & 2\hfill & -7\hfill \\ \hfill & \text{ }12\hfill & \text{ }\text{ }\text{ }22\hfill & 14\hfill & \text{ }\text{ }32\hfill \end{array}}\\ \begin{array}{llllll}\hfill & \text{}6\hfill & 11\hfill & \text{ }\text{ }\text{ }7\hfill & \text{ }\text{ }16\hfill & \text{ }\text{ }25\hfill \end{array}\end{array}[/latex]. Zero to 4 roots. You can also use the calculator to check your own manual math calculations to ensure your computations are correct and allow you to check any errors in your fourth degree equation calculation (s). of.the.function). There are four possibilities, as we can see below. [latex]\frac{p}{q}=\frac{\text{Factors of the constant term}}{\text{Factors of the leading coefficient}}=\pm 1,\pm 2,\pm 4,\pm \frac{1}{2}[/latex]. So for your set of given zeros, write: (x - 2) = 0. Because [latex]x=i[/latex]is a zero, by the Complex Conjugate Theorem [latex]x=-i[/latex]is also a zero. There are many ways to improve your writing skills, but one of the most effective is to practice writing regularly.
Maximum and Minimum Values of Polynomials - AlgebraLAB: Making Math and Similarly, if [latex]x-k[/latex]is a factor of [latex]f\left(x\right)[/latex],then the remainder of the Division Algorithm [latex]f\left(x\right)=\left(x-k\right)q\left(x\right)+r[/latex]is 0. This process assumes that all the zeroes are real numbers. Any help would be, Find length and width of rectangle given area, How to determine the parent function of a graph, How to find answers to math word problems, How to find least common denominator of rational expressions, Independent practice lesson 7 compute with scientific notation, Perimeter and area of a rectangle formula, Solving pythagorean theorem word problems. Begin by writing an equation for the volume of the cake. The zeros of [latex]f\left(x\right)[/latex]are 3 and [latex]\pm \frac{i\sqrt{3}}{3}[/latex]. For example, notice that the graph of f (x)= (x-1) (x-4)^2 f (x) = (x 1)(x 4)2 behaves differently around the zero 1 1 than around the zero 4 4, which is a double zero. find a formula for a fourth degree polynomial. A quartic function is a fourth-degree polynomial: a function which has, as its highest order term, a variable raised to the fourth power. We were given that the length must be four inches longer than the width, so we can express the length of the cake as [latex]l=w+4[/latex]. The Rational Zero Theorem tells us that if [latex]\frac{p}{q}[/latex] is a zero of [latex]f\left(x\right)[/latex],then pis a factor of 1 and qis a factor of 2. We can then set the quadratic equal to 0 and solve to find the other zeros of the function. This is also a quadratic equation that can be solved without using a quadratic formula. Get detailed step-by-step answers Transcribed image text: Find a fourth-degree polynomial function f(x) with real coefficients that has -1, 1, and i as zeros and such that f(3) = 160. = x 2 - 2x - 15. Search our database of more than 200 calculators. Find a fourth-degree polynomial with integer coefficients that has zeros 2i and 1, with 1 a zero of multiplicity 2. So either the multiplicity of [latex]x=-3[/latex] is 1 and there are two complex solutions, which is what we found, or the multiplicity at [latex]x=-3[/latex] is three. The polynomial generator generates a polynomial from the roots introduced in the Roots field. Now we have to evaluate the polynomial at all these values: So the polynomial roots are: The examples are great and work.
Quartic Equation Solver - Had2Know We use cookies to improve your experience on our site and to show you relevant advertising. We can see from the graph that the function has 0 positive real roots and 2 negative real roots. If you're looking for academic help, our expert tutors can assist you with everything from homework to . Grade 3 math division word problems worksheets, How do you find the height of a rectangular prism, How to find a missing side of a right triangle using trig, Price elasticity of demand equation calculator, Solving quadratic equation with solver in excel. Roots =. Solving math equations can be tricky, but with a little practice, anyone can do it! It will have at least one complex zero, call it [latex]{c}_{\text{2}}[/latex]. I designed this website and wrote all the calculators, lessons, and formulas. Welcome to MathPortal.
5.3 Graphs of Polynomial Functions - OpenStax [latex]\begin{array}{l}\frac{p}{q}=\frac{\text{Factors of the constant term}}{\text{Factor of the leading coefficient}}\hfill \\ \text{}\frac{p}{q}=\frac{\text{Factors of 3}}{\text{Factors of 3}}\hfill \end{array}[/latex]. Factor it and set each factor to zero. Since a fourth degree polynomial can have at most four zeros, including multiplicities, then the intercept x = -1 must only have multiplicity 2, which we had found through division, and not 3 as we had guessed. Use the Rational Zero Theorem to find the rational zeros of [latex]f\left(x\right)={x}^{3}-3{x}^{2}-6x+8[/latex]. Our full solution gives you everything you need to get the job done right.
Taylor Series Calculator | Instant Solutions - Voovers Use synthetic division to find the zeros of a polynomial function. Solving the equations is easiest done by synthetic division. Get support from expert teachers. The remainder is the value [latex]f\left(k\right)[/latex]. Ex: Polynomial Root of t^2+5t+6 Polynomial Root of -16t^2+24t+6 Polynomial Root of -16t^2+29t-12 Polynomial Root Calculator: Calculate Thus, the zeros of the function are at the point . (i) Here, + = and . = - 1. 4. of.the.function). This is the standard form of a quadratic equation, Example 01: Solve the equation $ 2x^2 + 3x - 14 = 0 $. [latex]\begin{array}{l}\frac{p}{q}=\frac{\text{Factors of the constant term}}{\text{Factors of the leading coefficient}}\hfill \\ \text{}\frac{p}{q}=\frac{\text{Factors of 1}}{\text{Factors of 2}}\hfill \end{array}[/latex]. The roots of the function are given as: x = + 2 x = - 2 x = + 2i x = - 2i Example 4: Find the zeros of the following polynomial function: f ( x) = x 4 - 4 x 2 + 8 x + 35 If the remainder is not zero, discard the candidate. The Linear Factorization Theorem tells us that a polynomial function will have the same number of factors as its degree, and each factor will be of the form (xc) where cis a complex number.
Use Descartes Rule of Signs to determine the maximum possible number of positive and negative real zeros for [latex]f\left(x\right)=2{x}^{4}-10{x}^{3}+11{x}^{2}-15x+12[/latex]. They want the length of the cake to be four inches longer than the width of the cake and the height of the cake to be one-third of the width. Did not begin to use formulas Ferrari - not interestingly. It also displays the step-by-step solution with a detailed explanation. This is true because any factor other than [latex]x-\left(a-bi\right)[/latex],when multiplied by [latex]x-\left(a+bi\right)[/latex],will leave imaginary components in the product. Dividing by [latex]\left(x - 1\right)[/latex]gives a remainder of 0, so 1 is a zero of the function. Either way, our result is correct. Each rational zero of a polynomial function with integer coefficients will be equal to a factor of the constant term divided by a factor of the leading coefficient. Recall that the Division Algorithm tells us [latex]f\left(x\right)=\left(x-k\right)q\left(x\right)+r[/latex]. Yes. [latex]f\left(x\right)[/latex]can be written as [latex]\left(x - 1\right){\left(2x+1\right)}^{2}[/latex]. The client tells the manufacturer that, because of the contents, the length of the container must be one meter longer than the width, and the height must be one meter greater than twice the width. of.the.function). It is interesting to note that we could greatly improve on the graph of y = f(x) in the previous example given to us by the calculator. Example 03: Solve equation $ 2x^2 - 10 = 0 $. Share Cite Follow You can also use the calculator to check your own manual math calculations to ensure your computations are correct and allow you to check any errors in your fourth degree equation calculation(s).
Calculator to find degree online - Solumaths Work on the task that is interesting to you. Every polynomial function with degree greater than 0 has at least one complex zero. (Use x for the variable.) Enter the equation in the fourth degree equation. Zero, one or two inflection points. What is polynomial equation? Use synthetic division to divide the polynomial by [latex]\left(x-k\right)[/latex]. By the Factor Theorem, the zeros of [latex]{x}^{3}-6{x}^{2}-x+30[/latex] are 2, 3, and 5. We already know that 1 is a zero. To solve a polynomial equation write it in standard form (variables and canstants on one side and zero on the other side of the equation). By the fundamental Theorem of Algebra, any polynomial of degree 4 can be Where, ,,, are the roots (or zeros) of the equation P(x)=0. Find a polynomial that has zeros $ 4, -2 $. This website's owner is mathematician Milo Petrovi. Ex: when I take a picture of let's say -6x-(-2x) I want to be able to tell the calculator to solve for the difference or the sum of that equations, the ads are nearly there too, it's in any language, and so easy to use, this app it great, it helps me work out problems for me to understand instead of just goveing me an answer. Factorized it is written as (x+2)*x*(x-3)*(x-4)*(x-5).
Find a Polynomial Given its Graph Questions with Solutions Find a fourth-degree polynomial with - Softmath The multiplicity of a zero is important because it tells us how the graph of the polynomial will behave around the zero. Continue to apply the Fundamental Theorem of Algebra until all of the zeros are found. The Rational Zero Theorem tells us that the possible rational zeros are [latex]\pm 3,\pm 9,\pm 13,\pm 27,\pm 39,\pm 81,\pm 117,\pm 351[/latex],and [latex]\pm 1053[/latex]. You can try first finding the rational roots using the rational root theorem in combination with the factor theorem in order to reduce the degree of the polynomial until you get to a quadratic, which can be solved by means of the quadratic formula or by completing the square. [latex]\begin{array}{l}f\left(x\right)=a\left(x+3\right)\left(x - 2\right)\left(x-i\right)\left(x+i\right)\\ f\left(x\right)=a\left({x}^{2}+x - 6\right)\left({x}^{2}+1\right)\\ f\left(x\right)=a\left({x}^{4}+{x}^{3}-5{x}^{2}+x - 6\right)\end{array}[/latex]. It is helpful for learning math better and easier than how it is usually taught, this app is so amazing, it takes me five minutes to do a whole page I just love it. Once we have done this, we can use synthetic division repeatedly to determine all of the zeros of a polynomial function. The number of negative real zeros is either equal to the number of sign changes of [latex]f\left(-x\right)[/latex] or is less than the number of sign changes by an even integer. I haven't met any app with such functionality and no ads and pays. Sol. A General Note: The Factor Theorem According to the Factor Theorem, k is a zero of [latex]f\left(x\right)[/latex] if and only if [latex]\left(x-k\right)[/latex] is a factor of [latex]f\left(x\right)[/latex]. In this example, the last number is -6 so our guesses are. Since polynomial with real coefficients. We will be discussing how to Find the fourth degree polynomial function with zeros calculator in this blog post. The sheet cake pan should have dimensions 13 inches by 9 inches by 3 inches. According to the Factor Theorem, kis a zero of [latex]f\left(x\right)[/latex]if and only if [latex]\left(x-k\right)[/latex]is a factor of [latex]f\left(x\right)[/latex]. If kis a zero, then the remainder ris [latex]f\left(k\right)=0[/latex]and [latex]f\left(x\right)=\left(x-k\right)q\left(x\right)+0[/latex]or [latex]f\left(x\right)=\left(x-k\right)q\left(x\right)[/latex]. Get the best Homework answers from top Homework helpers in the field. This is called the Complex Conjugate Theorem. This theorem forms the foundation for solving polynomial equations. Max/min of polynomials of degree 2: is a parabola and its graph opens upward from the vertex. f(x)=x^4+5x^2-36 If f(x) has zeroes at 2 and -2 it will have (x-2)(x+2) as factors. Given that,f (x) be a 4-th degree polynomial with real coefficients such that 3,-3,i as roots also f (2)=-50. Zero, one or two inflection points. By the Zero Product Property, if one of the factors of
I designed this website and wrote all the calculators, lessons, and formulas. Examine the behavior of the graph at the x -intercepts to determine the multiplicity of each factor. computer aided manufacturing the endmill cutter, The Definition of Monomials and Polynomials Video Tutorial, Math: Polynomials Tutorials and Revision Guides, The Definition of Monomials and Polynomials Revision Notes, Operations with Polynomials Revision Notes, Solutions for Polynomial Equations Revision Notes, Solutions for Polynomial Equations Practice Questions, Operations with Polynomials Practice Questions, The 4th Degree Equation Calculator will calculate the roots of the 4th degree equation you have entered. Each factor will be in the form [latex]\left(x-c\right)[/latex] where. This is what your synthetic division should have looked like: Note: there was no [latex]x[/latex] term, so a zero was needed, Another use for the Remainder Theorem is to test whether a rational number is a zero for a given polynomial, but first we need a pool of rational numbers to test. The calculator generates polynomial with given roots. There must be 4, 2, or 0 positive real roots and 0 negative real roots. This is the essence of the Rational Zero Theorem; it is a means to give us a pool of possible rational zeros. All the zeros can be found by setting each factor to zero and solving The factor x2 = x x which when set to zero produces two identical solutions, x = 0 and x = 0 The factor (x2 3x) = x(x 3) when set to zero produces two solutions, x = 0 and x = 3 Begin by determining the number of sign changes. This page includes an online 4th degree equation calculator that you can use from your mobile, device, desktop or tablet and also includes a supporting guide and instructions on how to use the calculator. The scaning works well too. The polynomial can be up to fifth degree, so have five zeros at maximum. According to the rule of thumbs: zero refers to a function (such as a polynomial), and the root refers to an equation. Further polynomials with the same zeros can be found by multiplying the simplest polynomial with a factor. Learn more Support us Since we are looking for a degree 4 polynomial and now have four zeros, we have all four factors. This calculator allows to calculate roots of any polynom of the fourth degree. Free time to spend with your family and friends. Use any other point on the graph (the y -intercept may be easiest) to determine the stretch factor. Math can be tough to wrap your head around, but with a little practice, it can be a breeze! [latex]-2, 1, \text{and } 4[/latex] are zeros of the polynomial. The last equation actually has two solutions. We found that both iand i were zeros, but only one of these zeros needed to be given.
Find the fourth degree polynomial function with zeros calculator Since 1 is not a solution, we will check [latex]x=3[/latex]. Just enter the expression in the input field and click on the calculate button to get the degree value along with show work. 2. powered by. Roots =. 2. Polynomial equations model many real-world scenarios. Find the zeros of [latex]f\left(x\right)=2{x}^{3}+5{x}^{2}-11x+4[/latex]. If you're struggling with a math problem, scanning it for key information can help you solve it more quickly. You can calculate the root of the fourth degree manually using the fourth degree equation below or you can use the fourth degree equation calculator and save yourself the time and hassle of calculating the math manually. Finding a Polynomial: Without Non-zero Points Example Find a polynomial of degree 4 with zeroes of -3 and 6 (multiplicity 3) Step 1: Set up your factored form: {eq}P (x) = a (x-z_1). An 4th degree polynominals divide calcalution. In this section, we will discuss a variety of tools for writing polynomial functions and solving polynomial equations. Input the roots here, separated by comma. 4 procedure of obtaining a factor and a quotient with degree 1 less than the previous. [latex]\begin{array}{l}V=\left(w+4\right)\left(w\right)\left(\frac{1}{3}w\right)\\ V=\frac{1}{3}{w}^{3}+\frac{4}{3}{w}^{2}\end{array}[/latex].
Quartic Function / Curve: Definition, Examples - Statistics How To Find the fourth degree polynomial function with zeros calculator First of all I like that you can take a picture of your problem and It can recognize it for you, but most of all how it explains the problem step by step, instead of just giving you the answer. Answer only. (Remember we were told the polynomial was of degree 4 and has no imaginary components). The factors of 1 are [latex]\pm 1[/latex] and the factors of 2 are [latex]\pm 1[/latex] and [latex]\pm 2[/latex]. Solving matrix characteristic equation for Principal Component Analysis. We will use synthetic division to evaluate each possible zero until we find one that gives a remainder of 0. at [latex]x=-3[/latex].
Find the fourth degree polynomial with zeros calculator The missing one is probably imaginary also, (1 +3i). We name polynomials according to their degree. The degree is the largest exponent in the polynomial. Other than that I love that it goes step by step so I can actually learn via reverse engineering, i found math app to be a perfect tool to help get me through my college algebra class, used by students who SHOULDNT USE IT and tutors like me WHO SHOULDNT NEED IT. Algebra Polynomial Division Calculator Step 1: Enter the expression you want to divide into the editor. Next, we examine [latex]f\left(-x\right)[/latex] to determine the number of negative real roots. Thanks for reading my bad writings, very useful. The 4th Degree Equation calculator Is an online math calculator developed by calculator to support with the development of your mathematical knowledge. Reference: If 2 + 3iwere given as a zero of a polynomial with real coefficients, would 2 3ialso need to be a zero? Use synthetic division to divide the polynomial by [latex]x-k[/latex]. If there are any complex zeroes then this process may miss some pretty important features of the graph. [latex]\begin{array}{l}\frac{p}{q}=\pm \frac{1}{1},\pm \frac{1}{2}\text{ }& \frac{p}{q}=\pm \frac{2}{1},\pm \frac{2}{2}\text{ }& \frac{p}{q}=\pm \frac{4}{1},\pm \frac{4}{2}\end{array}[/latex]. Enter the equation in the fourth degree equation 4 by 4 cube solver Best star wars trivia game Equation for perimeter of a rectangle Fastest way to solve 3x3 Function table calculator 3 variables How many liters are in 64 oz How to calculate . http://cnx.org/contents/9b08c294-057f-4201-9f48-5d6ad992740d@5.2. Hence complex conjugate of i is also a root. Also note the presence of the two turning points. Function's variable: Examples. Math is the study of numbers, space, and structure. Use the Linear Factorization Theorem to find polynomials with given zeros.
Solved Find a fourth degree polynomial function f(x) with | Chegg.com Substitute [latex]x=-2[/latex] and [latex]f\left(2\right)=100[/latex] If you're looking for support from expert teachers, you've come to the right place. The series will be most accurate near the centering point. Are zeros and roots the same?
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4th Degree Polynomials Division Calculation - MYMATHTABLES.COM The Factor Theorem is another theorem that helps us analyze polynomial equations.
Solving Quartic, or 4th Degree, Equations - Study.com In this case we divide $ 2x^3 - x^2 - 3x - 6 $ by $ \color{red}{x - 2}$. Find the zeros of [latex]f\left(x\right)=3{x}^{3}+9{x}^{2}+x+3[/latex]. Synthetic division can be used to find the zeros of a polynomial function.
How to Solve Polynomial Equations - brownmath.com The calculator generates polynomial with given roots. Now we use $ 2x^2 - 3 $ to find remaining roots. Try It #1 Find the y - and x -intercepts of the function f(x) = x4 19x2 + 30x. If you need help, don't hesitate to ask for it. . Fourth Degree Polynomial Equations Formula y = ax 4 + bx 3 + cx 2 + dx + e 4th degree polynomials are also known as quartic polynomials. Evaluate a polynomial using the Remainder Theorem. the degree of polynomial $ p(x) = 8x^\color{red}{2} + 3x -1 $ is $\color{red}{2}$. One way to ensure that math tasks are clear is to have students work in pairs or small groups to complete the task. In this case we have $ a = 2, b = 3 , c = -14 $, so the roots are: Sometimes, it is much easier not to use a formula for finding the roots of a quadratic equation. This polynomial graphing calculator evaluates one-variable polynomial functions up to the fourth-order, for given coefficients. Use the Remainder Theorem to evaluate [latex]f\left(x\right)=6{x}^{4}-{x}^{3}-15{x}^{2}+2x - 7[/latex]at [latex]x=2[/latex]. A non-polynomial function or expression is one that cannot be written as a polynomial. According to the Fundamental Theorem of Algebra, every polynomial function has at least one complex zero. Notice that two of the factors of the constant term, 6, are the two numerators from the original rational roots: 2 and 3. We will use synthetic division to evaluate each possible zero until we find one that gives a remainder of 0. Lets write the volume of the cake in terms of width of the cake. Determine which possible zeros are actual zeros by evaluating each case of [latex]f\left(\frac{p}{q}\right)[/latex]. Statistics: 4th Order Polynomial. Once the polynomial has been completely factored, we can easily determine the zeros of the polynomial. Coefficients can be both real and complex numbers. At [latex]x=1[/latex], the graph crosses the x-axis, indicating the odd multiplicity (1,3,5) for the zero [latex]x=1[/latex]. The calculator generates polynomial with given roots. We can use synthetic division to show that [latex]\left(x+2\right)[/latex] is a factor of the polynomial. Polynomial From Roots Generator input roots 1/2,4 and calculator will generate a polynomial show help examples Enter roots: display polynomial graph Generate Polynomial examples example 1: Answer provided by our tutors the 4-degree polynomial with integer coefficients that has zeros 2i and 1, with 1 a zero of multiplicity 2 the zeros are 2i, -2i, -1, and -1 We offer fast professional tutoring services to help improve your grades. 1 is the only rational zero of [latex]f\left(x\right)[/latex]. Non-polynomial functions include trigonometric functions, exponential functions, logarithmic functions, root functions, and more. The calculator is also able to calculate the degree of a polynomial that uses letters as coefficients. Find zeros of the function: f x 3 x 2 7 x 20. Step 1/1. We can infer that the numerators of the rational roots will always be factors of the constant term and the denominators will be factors of the leading coefficient. For example, View the full answer. To solve the math question, you will need to first figure out what the question is asking. For the given zero 3i we know that -3i is also a zero since complex roots occur in. If any of the four real zeros are rational zeros, then they will be of one of the following factors of 4 divided by one of the factors of 2. Can't believe this is free it's worthmoney. Function zeros calculator. The calculator generates polynomial with given roots. Please tell me how can I make this better. I really need help with this problem.
Find a degree 3 polynomial with zeros calculator | Math Index Thus, all the x-intercepts for the function are shown. This calculator allows to calculate roots of any polynom of the fourth degree. A new bakery offers decorated sheet cakes for childrens birthday parties and other special occasions. 4th Degree Equation Solver. Calculator shows detailed step-by-step explanation on how to solve the problem. Find the zeros of [latex]f\left(x\right)=4{x}^{3}-3x - 1[/latex]. Identifying Zeros and Their Multiplicities Graphs behave differently at various x -intercepts.
Polynomial Roots Calculator that shows work - MathPortal Find a third degree polynomial with real coefficients that has zeros of 5 and 2isuch that [latex]f\left(1\right)=10[/latex]. Math equations are a necessary evil in many people's lives. Now we can split our equation into two, which are much easier to solve. The possible values for [latex]\frac{p}{q}[/latex] are [latex]\pm 1[/latex] and [latex]\pm \frac{1}{2}[/latex]. Step 3: If any zeros have a multiplicity other than 1, set the exponent of the matching factor to the given multiplicity.
If the polynomial is divided by x k, the remainder may be found quickly by evaluating the polynomial function at k, that is, f(k).