find the fourth degree polynomial with zeros calculator

The remainder is zero, so [latex]\left(x+2\right)[/latex] is a factor of the polynomial. I love spending time with my family and friends. Calculator shows detailed step-by-step explanation on how to solve the problem. Step 2: Click the blue arrow to submit and see the result! This means that we can factor the polynomial function into nfactors. Only multiplication with conjugate pairs will eliminate the imaginary parts and result in real coefficients. To solve the math question, you will need to first figure out what the question is asking. The Rational Zero Theorem helps us to narrow down the number of possible rational zeros using the ratio of the factors of the constant term and factors of the leading coefficient of the polynomial. math is the study of numbers, shapes, and patterns. It's an amazing app! (xr) is a factor if and only if r is a root. The bakery wants the volume of a small cake to be 351 cubic inches. To solve a math equation, you need to decide what operation to perform on each side of the equation. 4 procedure of obtaining a factor and a quotient with degree 1 less than the previous. Because [latex]x=i[/latex]is a zero, by the Complex Conjugate Theorem [latex]x=-i[/latex]is also a zero. 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. Substitute [latex]\left(c,f\left(c\right)\right)[/latex] into the function to determine the leading coefficient. I haven't met any app with such functionality and no ads and pays. Factor it and set each factor to zero. Examine the behavior of the graph at the x -intercepts to determine the multiplicity of each factor. Find a fourth-degree polynomial with integer coefficients that has zeros 2i and 1, with 1 a zero of multiplicity 2. This is the essence of the Rational Zero Theorem; it is a means to give us a pool of possible rational zeros. 2. Use the factors to determine the zeros of the polynomial. According to Descartes Rule of Signs, if we let [latex]f\left(x\right)={a}_{n}{x}^{n}+{a}_{n - 1}{x}^{n - 1}++{a}_{1}x+{a}_{0}[/latex]be a polynomial function with real coefficients: Use Descartes Rule of Signs to determine the possible numbers of positive and negative real zeros for [latex]f\left(x\right)=-{x}^{4}-3{x}^{3}+6{x}^{2}-4x - 12[/latex]. Just enter the expression in the input field and click on the calculate button to get the degree value along with show work. [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]. [latex]\begin{array}{l}\text{ }351=\frac{1}{3}{w}^{3}+\frac{4}{3}{w}^{2}\hfill & \text{Substitute 351 for }V.\hfill \\ 1053={w}^{3}+4{w}^{2}\hfill & \text{Multiply both sides by 3}.\hfill \\ \text{ }0={w}^{3}+4{w}^{2}-1053 \hfill & \text{Subtract 1053 from both sides}.\hfill \end{array}[/latex]. Input the roots here, separated by comma. The calculator computes exact solutions for quadratic, cubic, and quartic equations. This website's owner is mathematician Milo Petrovi. Use synthetic division to find the zeros of a polynomial function. If there are any complex zeroes then this process may miss some pretty important features of the graph. 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. The Rational Zero Theorem states that if the polynomial [latex]f\left(x\right)={a}_{n}{x}^{n}+{a}_{n - 1}{x}^{n - 1}++{a}_{1}x+{a}_{0}[/latex] has integer coefficients, then every rational zero of [latex]f\left(x\right)[/latex]has the form [latex]\frac{p}{q}[/latex] where pis a factor of the constant term [latex]{a}_{0}[/latex] and qis a factor of the leading coefficient [latex]{a}_{n}[/latex]. Roots =. 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 quadratic is a perfect square. Left no crumbs and just ate . Find more Mathematics widgets in Wolfram|Alpha. To answer this question, I have to remember that the polynomial's degree gives me the ceiling on the number of bumps. 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]. When any complex number with an imaginary component is given as a zero of a polynomial with real coefficients, the conjugate must also be a zero of the polynomial. 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]. Quartic Equation Solver & Quartic Formula Fourth-degree polynomials, equations of the form Ax4 + Bx3 + Cx2 + Dx + E = 0 where A is not equal to zero, are called quartic equations. 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. [latex]f\left(x\right)=-\frac{1}{2}{x}^{3}+\frac{5}{2}{x}^{2}-2x+10[/latex]. 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. We can use the Factor Theorem to completely factor a polynomial into the product of nfactors. Use synthetic division to check [latex]x=1[/latex]. Lets begin with 3. 4th degree: Quartic equation solution Use numeric methods If the polynomial degree is 5 or higher Isolate the root bounds by VAS-CF algorithm: Polynomial root isolation. Because the graph crosses the x axis at x = 0 and x = 5 / 2, both zero have an odd multiplicity. The polynomial generator generates a polynomial from the roots introduced in the Roots field. Create the term of the simplest polynomial from the given zeros. It is called the zero polynomial and have no degree. Next, we examine [latex]f\left(-x\right)[/latex] to determine the number of negative real roots. For any root or zero of a polynomial, the relation (x - root) = 0 must hold by definition of a root: where the polynomial crosses zero. Polynomial Degree Calculator Find the degree of a polynomial function step-by-step full pad Examples A polynomial is an expression of two or more algebraic terms, often having different exponents. Thanks for reading my bad writings, very useful. 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 andqis a factor of 4. This theorem forms the foundation for solving polynomial equations. If the remainder is 0, the candidate is a zero. The calculator generates polynomial with given roots. This website's owner is mathematician Milo Petrovi. Again, there are two sign changes, so there are either 2 or 0 negative real roots. If the polynomial function fhas real coefficients and a complex zero of the form [latex]a+bi[/latex],then the complex conjugate of the zero, [latex]a-bi[/latex],is also a zero. They can also be useful for calculating ratios. [emailprotected], find real and complex zeros of a polynomial, find roots of the polynomial $4x^2 - 10x + 4$, find polynomial roots $-2x^4 - x^3 + 189$, solve equation $6x^3 - 25x^2 + 2x + 8 = 0$, Search our database of more than 200 calculators. 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. I designed this website and wrote all the calculators, lessons, and formulas. Find a fourth Find a fourth-degree polynomial function with zeros 1, -1, i, -i. Once you understand what the question is asking, you will be able to solve it. We can use the relationships between the width and the other dimensions to determine the length and height of the sheet cake pan. Coefficients can be both real and complex numbers. Use any other point on the graph (the y -intercept may be easiest) to determine the stretch factor. According to the Linear Factorization Theorem, a polynomial function will have the same number of factors as its degree, and each factor will be of the form [latex]\left(x-c\right)[/latex] where cis a complex number. Zeros: Notation: xn or x^n Polynomial: Factorization: Finding the x -Intercepts of a Polynomial Function Using a Graph Find the x -intercepts of h(x) = x3 + 4x2 + x 6. Statistics: 4th Order Polynomial. In most real-life applications, we use polynomial regression of rather low degrees: Degree 1: y = a0 + a1x As we've already mentioned, this is simple linear regression, where we try to fit a straight line to the data points. Purpose of use. Which polynomial has a double zero of $5$ and has $\frac{2}{3}$ as a simple zero? To solve a cubic equation, the best strategy is to guess one of three roots. The possible values for [latex]\frac{p}{q}[/latex] are [latex]\pm 1,\pm \frac{1}{2}[/latex], and [latex]\pm \frac{1}{4}[/latex]. x4+. According to the Fundamental Theorem of Algebra, every polynomial function has at least one complex zero. The graph is shown at right using the WINDOW (-5, 5) X (-2, 16). If possible, continue until the quotient is a quadratic. Thus the polynomial formed. The Fundamental Theorem of Algebra tells us that every polynomial function has at least one complex zero. Question: Find the fourth-degree polynomial function with zeros 4, -4 , 4i , and -4i. You can track your progress on your fitness journey by recording your workouts, monitoring your food intake, and taking note of any changes in your body. So for your set of given zeros, write: (x - 2) = 0. One way to ensure that math tasks are clear is to have students work in pairs or small groups to complete the task. The degree is the largest exponent in the polynomial. For example, the degree of polynomial p(x) = 8x2 + 3x 1 is 2. The missing one is probably imaginary also, (1 +3i). 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. Did not begin to use formulas Ferrari - not interestingly. Example 3: Find a quadratic polynomial whose sum of zeros and product of zeros are respectively , - 1. Find a Polynomial Function Given the Zeros and. 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]. quadratic - degree 2, Cubic - degree 3, and Quartic - degree 4. Here is the online 4th degree equation solver for you to find the roots of the fourth-degree equations. For the given zero 3i we know that -3i is also a zero since complex roots occur in We can see from the graph that the function has 0 positive real roots and 2 negative real roots. If you need your order fast, we can deliver it to you in record time. This means that, since there is a 3rd degree polynomial, we are looking at the maximum number of turning points. Calculator shows detailed step-by-step explanation on how to solve the problem. Install calculator on your site. 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. A vital implication of the Fundamental Theorem of Algebrais that a polynomial function of degree nwill have nzeros in the set of complex numbers if we allow for multiplicities. 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]. (where "z" is the constant at the end): z/a (for even degree polynomials like quadratics) z/a (for odd degree polynomials like cubics) It works on Linear, Quadratic, Cubic and Higher! It will have at least one complex zero, call it [latex]{c}_{\text{2}}[/latex]. Get the best Homework answers from top Homework helpers in the field. Repeat step two using the quotient found from synthetic division. P(x) = A(x^2-11)(x^2+4) Where A is an arbitrary integer. Polynomial Functions of 4th Degree. 2. powered by. Evaluate a polynomial using the Remainder Theorem. For us, the most interesting ones are: We can use this theorem to argue that, if [latex]f\left(x\right)[/latex] is a polynomial of degree [latex]n>0[/latex], and ais a non-zero real number, then [latex]f\left(x\right)[/latex] has exactly nlinear factors. Lets begin with 1. It is used in everyday life, from counting to measuring to more complex calculations. At 24/7 Customer Support, we are always here to help you with whatever you need. Get support from expert teachers. If you need an answer fast, you can always count on Google. The Rational Zero Theorem helps us to narrow down the list of possible rational zeros for a polynomial function. In this case, a = 3 and b = -1 which gives . Solution Because x = i x = i is a zero, by the Complex Conjugate Theorem x = - i x = - i is also a zero. We have now introduced a variety of tools for solving polynomial equations. The zeros of the function are 1 and [latex]-\frac{1}{2}[/latex] with multiplicity 2. It also displays the step-by-step solution with a detailed explanation. Finding polynomials with given zeros and degree calculator - This video will show an example of solving a polynomial equation using a calculator. The Fundamental Theorem of Algebra states that, if [latex]f(x)[/latex] is a polynomial of degree [latex]n>0[/latex], then [latex]f(x)[/latex] has at least one complex zero. If 2 + 3iwere given as a zero of a polynomial with real coefficients, would 2 3ialso need to be a zero? Find the remaining factors. This step-by-step guide will show you how to easily learn the basics of HTML. This free math tool finds the roots (zeros) of a given polynomial. Mathematics is a way of dealing with tasks that involves numbers and equations. This is also a quadratic equation that can be solved without using a quadratic formula. Max/min of polynomials of degree 2: is a parabola and its graph opens upward from the vertex. The Factor Theorem is another theorem that helps us analyze polynomial equations. Graphing calculators can be used to find the real, if not rational, solutions, of quartic functions. 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. Use the Fundamental Theorem of Algebra to find complex zeros of a polynomial function. Amazing, And Super Helpful for Math brain hurting homework or time-taking assignments, i'm quarantined, that's bad enough, I ain't doing math, i haven't found a math problem that it hasn't solved. 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. So, the end behavior of increasing without bound to the right and decreasing without bound to the left will continue. This is called the Complex Conjugate Theorem. Allowing for multiplicities, a polynomial function will have the same number of factors as its degree. Enter values for a, b, c and d and solutions for x will be calculated. Calculator shows detailed step-by-step explanation on how to solve the problem. Math equations are a necessary evil in many people's lives. The eleventh-degree polynomial (x + 3) 4 (x 2) 7 has the same zeroes as did the quadratic, but in this case, the x = 3 solution has multiplicity 4 because the factor (x + 3) occurs four times (that is, the factor is raised to the fourth power) and the x = 2 solution has multiplicity 7 because the factor (x 2) occurs seven times. Once the polynomial has been completely factored, we can easily determine the zeros of the polynomial. The calculator generates polynomial with given roots. Identifying Zeros and Their Multiplicities Graphs behave differently at various x -intercepts. 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]. Really good app for parents, students and teachers to use to check their math work. The number of negative real zeros of a polynomial function is either the number of sign changes of [latex]f\left(-x\right)[/latex] or less than the number of sign changes by an even integer. This is really appreciated . Also note the presence of the two turning points. 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 3 andqis a factor of 3. Zero to 4 roots. The polynomial can be written as [latex]\left(x - 1\right)\left(4{x}^{2}+4x+1\right)[/latex]. Find a third degree polynomial with real coefficients that has zeros of 5 and 2isuch that [latex]f\left(1\right)=10[/latex]. If you're struggling with math, there are some simple steps you can take to clear up the confusion and start getting the right answers. Math problems can be determined by using a variety of methods. Quartic Polynomials Division Calculator. Because our equation now only has two terms, we can apply factoring. 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: Zero, one or two inflection points. [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]. 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. There are many ways to improve your writing skills, but one of the most effective is to practice writing regularly. Now we have to divide polynomial with $ \color{red}{x - \text{ROOT}} $. Hence complex conjugate of i is also a root. These are the possible rational zeros for the function. In this case we divide $ 2x^3 - x^2 - 3x - 6 $ by $ \color{red}{x - 2}$. 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). [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]. I would really like it if the "why" button was free but overall I think it's great for anyone who is struggling in math or simply wants to check their answers. Once we have done this, we can use synthetic division repeatedly to determine all of the zeros of a polynomial function. We name polynomials according to their degree. The polynomial can be up to fifth degree, so have five zeros at maximum. 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). You may also find the following Math calculators useful. Therefore, [latex]f\left(x\right)[/latex] has nroots if we allow for multiplicities. Yes. Use the Rational Zero Theorem to list all possible rational zeros of the function. (x - 1 + 3i) = 0. Example 1 Sketch the graph of P (x) =5x5 20x4+5x3+50x2 20x 40 P ( x) = 5 x 5 20 x 4 + 5 x 3 + 50 x 2 20 x 40 . 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. Determine all factors of the constant term and all factors of the leading coefficient. To find the remainder using the Remainder Theorem, use synthetic division to divide the polynomial by [latex]x - 2[/latex]. There is a similar relationship between the number of sign changes in [latex]f\left(-x\right)[/latex] and the number of negative real 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 Find the roots in the positive field only if the input polynomial is even or odd (detected on 1st step) Algebra Polynomial Division Calculator Step 1: Enter the expression you want to divide into the editor. Calculus . I really need help with this problem. These are the possible rational zeros for the function. If you want to contact me, probably have some questions, write me using the contact form or email me on We will be discussing how to Find the fourth degree polynomial function with zeros calculator in this blog post. If you divide both sides of the equation by A you can simplify the equation to x4 + bx3 + cx2 + dx + e = 0. 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. Roots =. 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]. [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]. a 3, a 2, a 1 and a 0 are also constants, but they may be equal to zero. We can use synthetic division to show that [latex]\left(x+2\right)[/latex] is a factor of the polynomial. We name polynomials according to their degree. Use the Factor Theorem to solve a polynomial equation. All steps. (i) Here, + = and . = - 1. Continue to apply the Fundamental Theorem of Algebra until all of the zeros are found. Mathematics is a way of dealing with tasks that involves numbers and equations. A shipping container in the shape of a rectangular solid must have a volume of 84 cubic meters. We will use synthetic division to evaluate each possible zero until we find one that gives a remainder of 0. A non-polynomial function or expression is one that cannot be written as a polynomial. It has helped me a lot and it has helped me remember and it has also taught me things my teacher can't explain to my class right. Find the zeros of the quadratic function. Look at the graph of the function f. Notice that, at [latex]x=-3[/latex], the graph crosses the x-axis, indicating an odd multiplicity (1) for the zero [latex]x=-3[/latex]. Factoring 4th Degree Polynomials Example 2: Find all real zeros of the polynomial P(x) = 2x. Now that we can find rational zeros for a polynomial function, we will look at a theorem that discusses the number of complex zeros of a polynomial function. We offer fast professional tutoring services to help improve your grades. 1, 2 or 3 extrema. example. of.the.function). . 4th Degree Equation Solver. Our full solution gives you everything you need to get the job done right. Share Cite Follow 4. Step 3: If any zeros have a multiplicity other than 1, set the exponent of the matching factor to the given multiplicity. 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. The first one is obvious. If you need help, don't hesitate to ask for it. 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. Use the Linear Factorization Theorem to find polynomials with given zeros. Input the roots here, separated by comma. The multiplicity of a zero is important because it tells us how the graph of the polynomial will behave around the zero. We were given that the height of the cake is one-third of the width, so we can express the height of the cake as [latex]h=\frac{1}{3}w[/latex]. Select the zero option . [latex]\begin{array}{l}2x+1=0\hfill \\ \text{ }x=-\frac{1}{2}\hfill \end{array}[/latex]. Edit: Thank you for patching the camera. These x intercepts are the zeros of polynomial f (x). [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]. Get help from our expert homework writers! Welcome to MathPortal. We can write the polynomial quotient as a product of [latex]x-{c}_{\text{2}}[/latex] and a new polynomial quotient of degree two. Look at the graph of the function f. Notice, at [latex]x=-0.5[/latex], the graph bounces off the x-axis, indicating the even multiplicity (2,4,6) for the zero 0.5. = x 2 - (sum of zeros) x + Product of zeros. Now we use $ 2x^2 - 3 $ to find remaining roots. Solving math equations can be tricky, but with a little practice, anyone can do it! Untitled Graph. Quartics has the following characteristics 1. . About Press Copyright Contact us Creators Advertise Developers Terms Privacy Policy & Safety How YouTube works Test new features Press Copyright Contact us Creators . Adding polynomials. We can determine which of the possible zeros are actual zeros by substituting these values for xin [latex]f\left(x\right)[/latex]. Use synthetic division to evaluate a given possible zero by synthetically dividing the candidate into the polynomial. We found that both iand i were zeros, but only one of these zeros needed to be given. (I would add 1 or 3 or 5, etc, if I were going from the number . This process assumes that all the zeroes are real numbers. Recall that the Division Algorithm states that given a polynomial dividend f(x)and a non-zero polynomial divisor d(x)where the degree ofd(x) is less than or equal to the degree of f(x), there exist unique polynomials q(x)and r(x)such that, [latex]f\left(x\right)=d\left(x\right)q\left(x\right)+r\left(x\right)[/latex], If the divisor, d(x), is x k, this takes the form, [latex]f\left(x\right)=\left(x-k\right)q\left(x\right)+r[/latex], Since the divisor x kis linear, the remainder will be a constant, r. And, if we evaluate this for x =k, we have, [latex]\begin{array}{l}f\left(k\right)=\left(k-k\right)q\left(k\right)+r\hfill \\ \text{}f\left(k\right)=0\cdot q\left(k\right)+r\hfill \\ \text{}f\left(k\right)=r\hfill \end{array}[/latex]. Now we can split our equation into two, which are much easier to solve. Find a fourth degree polynomial with real coefficients that has zeros of -3, 2, i, i, such that f ( 2) = 100. f ( 2) = 100. The calculator generates polynomial with given roots. The polynomial must have factors of [latex]\left(x+3\right),\left(x - 2\right),\left(x-i\right)[/latex], and [latex]\left(x+i\right)[/latex].

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