For the given zero 3i we know that -3i is also a zero since complex roots occur in. The calculator computes exact solutions for quadratic, cubic, and quartic equations. Lets begin with 3. But this is for sure one, this app help me understand on how to solve question easily, this app is just great keep the good work! First we must find all the factors of the constant term, since the root of a polynomial is also a factor of its constant term. Get the best Homework answers from top Homework helpers in the field. The polynomial division calculator allows you to take a simple or complex expression and find the quotient and remainder instantly. Get detailed step-by-step answers Find zeros of the function: f x 3 x 2 7 x 20. 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. 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. Zeros: Notation: xn or x^n Polynomial: Factorization: Because our equation now only has two terms, we can apply factoring. The Polynomial Roots Calculator will display the roots of any polynomial with just one click after providing the input polynomial in the below input box and clicking on the calculate button. Calculator shows detailed step-by-step explanation on how to solve the problem. Loading. Synthetic division can be used to find the zeros of a polynomial function. Example 3: Find a quadratic polynomial whose sum of zeros and product of zeros are respectively , - 1. As we can see, a Taylor series may be infinitely long if we choose, but we may also . Algebra - Graphing Polynomials - Lamar University The factors of 1 are [latex]\pm 1[/latex] and the factors of 2 are [latex]\pm 1[/latex] and [latex]\pm 2[/latex]. P(x) = A(x^2-11)(x^2+4) Where A is an arbitrary integer. Factoring 4th Degree Polynomials Example 2: Find all real zeros of the polynomial P(x) = 2x. Enter the equation in the fourth degree equation. 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]. Again, there are two sign changes, so there are either 2 or 0 negative real roots. Solution Because x = i x = i is a zero, by the Complex Conjugate Theorem x = - i x = - i is also a zero. 4th Degree Equation Solver Here is the online 4th degree equation solver for you to find the roots of the fourth-degree equations. quadratic - degree 2, Cubic - degree 3, and Quartic - degree 4. x4+. Write the polynomial as the product of [latex]\left(x-k\right)[/latex] and the quadratic quotient. What is a fourth degree polynomial function with real coefficients that If possible, continue until the quotient is a quadratic. Left no crumbs and just ate . Quartic Equation Solver - Had2Know Our online calculator, based on Wolfram Alpha system is able to find zeros of almost any, even very complicated function. of.the.function). Roots =. 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]. Quartic Equation Calculation - MYMATHTABLES.COM If you're looking for support from expert teachers, you've come to the right place. All steps. Finding polynomials with given zeros and degree calculator - This video will show an example of solving a polynomial equation using a calculator. Use Descartes Rule of Signsto determine the maximum number of possible real zeros of a polynomial function. 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 calculator generates polynomial with given roots. Max/min of polynomials of degree 2: is a parabola and its graph opens upward from the vertex. Zeros Calculator + Online Solver With Free Steps - Story of Mathematics 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. Let's sketch a couple of polynomials. Zero, one or two inflection points. It's an amazing app! This theorem forms the foundation for solving polynomial equations. I am passionate about my career and enjoy helping others achieve their career goals. Hence the polynomial formed. 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. The polynomial generator generates a polynomial from the roots introduced in the Roots field. Get the free "Zeros Calculator" widget for your website, blog, Wordpress, Blogger, or iGoogle. According to the rule of thumbs: zero refers to a function (such as a polynomial), and the root refers to an equation. Find the fourth degree polynomial function with zeros calculator If you need help, our customer service team is available 24/7. 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. Use any other point on the graph (the y -intercept may be easiest) to determine the stretch factor. Calculator shows detailed step-by-step explanation on how to solve the problem. It . Degree of a Polynomial Calculator | Tool to Find Polynomial Degree Value Real numbers are also complex numbers. There is a straightforward way to determine the possible numbers of positive and negative real zeros for any polynomial function. This is the Factor Theorem: finding the roots or finding the factors is essentially the same thing. 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. Ex: Degree of a polynomial x^2+6xy+9y^2 Find a Polynomial Function Given the Zeros and. Once the polynomial has been completely factored, we can easily determine the zeros of the polynomial. 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. Now we have to evaluate the polynomial at all these values: So the polynomial roots are: 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. If you want to get the best homework answers, you need to ask the right questions. Wolfram|Alpha Widgets: "Zeros Calculator" - Free Mathematics Widget a 3, a 2, a 1 and a 0 are also constants, but they may be equal to zero. . 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]. The series will be most accurate near the centering point. math is the study of numbers, shapes, and patterns. Thus, all the x-intercepts for the function are shown. Lets walk through the proof of the theorem. Answer only. Factorized it is written as (x+2)*x*(x-3)*(x-4)*(x-5). 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. Share Cite Follow 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). 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]. The zeros of [latex]f\left(x\right)[/latex]are 3 and [latex]\pm \frac{i\sqrt{3}}{3}[/latex]. For example, This helps us to focus our resources and support current calculators and develop further math calculators to support our global community. Polynomial Roots Calculator that shows work - MathPortal 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). The solver will provide step-by-step instructions on how to Find the fourth degree polynomial function with zeros calculator. Repeat step two using the quotient found from synthetic division. The leading coefficient is 2; the factors of 2 are [latex]q=\pm 1,\pm 2[/latex]. Writing Formulas for Polynomial Functions | College Algebra We can use synthetic division to show that [latex]\left(x+2\right)[/latex] is a factor of the polynomial. We need to find a to ensure [latex]f\left(-2\right)=100[/latex]. Calculus . As we will soon see, a polynomial of degree nin the complex number system will have nzeros. If you divide both sides of the equation by A you can simplify the equation to x4 + bx3 + cx2 + dx + e = 0. Use a graph to verify the number of positive and negative real zeros for the function. (x - 1 + 3i) = 0. 3.6 Zeros of Polynomial Functions - Precalculus 2e - OpenStax 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 highest exponent is the order of the equation. It also displays the step-by-step solution with a detailed explanation. Quartics has the following characteristics 1. 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]. Fourth Degree Equation. The zeros of the function are 1 and [latex]-\frac{1}{2}[/latex] with multiplicity 2. [latex]\begin{array}{l}2x+1=0\hfill \\ \text{ }x=-\frac{1}{2}\hfill \end{array}[/latex]. For example within computer aided manufacturing the endmill cutter if often associated with the torus shape which requires the quartic solution in order to calculate its location relative to a triangulated surface. This calculator allows to calculate roots of any polynom of the fourth degree. The degree is the largest exponent in the polynomial. No. 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. Find the zeros of the quadratic function. There are four possibilities, as we can see below. 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. (adsbygoogle = window.adsbygoogle || []).push({}); If you found the Quartic Equation Calculator useful, it would be great if you would kindly provide a rating for the calculator and, if you have time, share to your favoursite social media netowrk. Math can be a difficult subject for some students, but with practice and persistence, anyone can master it. The possible values for [latex]\frac{p}{q}[/latex], and therefore the possible rational zeros for the function, are [latex]\pm 3, \pm 1, \text{and} \pm \frac{1}{3}[/latex]. These x intercepts are the zeros of polynomial f (x). Math is the study of numbers, space, and structure. The degree is the largest exponent in the polynomial. The Factor Theorem is another theorem that helps us analyze polynomial equations. [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]. Math can be tough to wrap your head around, but with a little practice, it can be a breeze! Polynomial Graphs: Zeroes and Their Multiplicities | Purplemath Are zeros and roots the same? 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. 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). Evaluate a polynomial using the Remainder Theorem. 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 We can conclude if kis a zero of [latex]f\left(x\right)[/latex], then [latex]x-k[/latex] is a factor of [latex]f\left(x\right)[/latex]. Zeros of a polynomial calculator - AtoZmath.com Zero to 4 roots. Question: Find the fourth-degree polynomial function with zeros 4, -4 , 4i , and -4i. If you want to contact me, probably have some questions, write me using the contact form or email me on Now we apply the Fundamental Theorem of Algebra to the third-degree polynomial quotient. 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. We already know that 1 is a zero. In the last section, we learned how to divide polynomials. Make Polynomial from Zeros - Rechneronline To solve a math equation, you need to decide what operation to perform on each side of the equation. We found that both iand i were zeros, but only one of these zeros needed to be given. Let the polynomial be ax 2 + bx + c and its zeros be and . Polynomial Functions of 4th Degree - Desmos | Let's learn together. Here is the online 4th degree equation solver for you to find the roots of the fourth-degree equations. The remainder is zero, so [latex]\left(x+2\right)[/latex] is a factor of the polynomial. Find a fourth degree polynomial with real coefficients that has zeros of -3, 2, i, i, such that f ( 2) = 100. f ( 2) = 100. Quartics has the following characteristics 1. There will be four of them and each one will yield a factor of [latex]f\left(x\right)[/latex]. So for your set of given zeros, write: (x - 2) = 0. Polynomial Division Calculator - Mathway Adding polynomials. For those who already know how to caluclate the Quartic Equation and want to save time or check their results, you can use the Quartic Equation Calculator by following the steps below: The Quartic Equation formula was first discovered by Lodovico Ferrari in 1540 all though it was claimed that in 1486 a Spanish mathematician was allegedly told by Toms de Torquemada, a Chief inquisitor of the Spanish Inquisition, that "it was the will of god that such a solution should be inaccessible to human understanding" which resulted in the mathematician being burned at the stake. There are many different forms that can be used to provide information. Two possible methods for solving quadratics are factoring and using the quadratic formula. By the Zero Product Property, if one of the factors of of.the.function). If f(x) has a zero at -3i then (x+3i) will be a factor and we will need to use a fourth factor to "clear" the imaginary component from the coefficients. The solutions are the solutions of the polynomial equation. Solving math equations can be challenging, but it's also a great way to improve your problem-solving skills. Degree 2: y = a0 + a1x + a2x2 Enter the equation in the fourth degree equation. [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]. Edit: Thank you for patching the camera. Determine all factors of the constant term and all factors of the leading coefficient. View the full answer. Here is the online 4th degree equation solver for you to find the roots of the fourth-degree equations. You can use it to help check homework questions and support your calculations of fourth-degree equations. To solve the math question, you will need to first figure out what the question is asking. Solving matrix characteristic equation for Principal Component Analysis. 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 polynomial can be up to fifth degree, so have five zeros at maximum. Factor it and set each factor to zero. Math problems can be determined by using a variety of methods. Find the fourth degree polynomial with zeros calculator | Math Index 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. Use synthetic division to divide the polynomial by [latex]x-k[/latex]. The Rational Zero Theorem helps us to narrow down the list of possible rational zeros for a polynomial function. (Remember we were told the polynomial was of degree 4 and has no imaginary components). 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. Find the roots in the positive field only if the input polynomial is even or odd (detected on 1st step) Let us set each factor equal to 0 and then construct the original quadratic function. Of course this vertex could also be found using the calculator. The best way to download full math explanation, it's download answer here. . This is the essence of the Rational Zero Theorem; it is a means to give us a pool of possible rational zeros. Finding roots of the fourth degree polynomial: $2x^4 + 3x^3 - 11x^2 [latex]\begin{array}{lll}f\left(x\right) & =6{x}^{4}-{x}^{3}-15{x}^{2}+2x - 7 \\ f\left(2\right) & =6{\left(2\right)}^{4}-{\left(2\right)}^{3}-15{\left(2\right)}^{2}+2\left(2\right)-7 \\ f\left(2\right) & =25\hfill \end{array}[/latex]. Show Solution. Use the Rational Zero Theorem to list all possible rational zeros of the function. This process assumes that all the zeroes are real numbers. Maximum and Minimum Values of Polynomials - AlgebraLAB: Making Math and Solving equations 4th degree polynomial equations - AbakBot-online Factor it and set each factor to zero. Polynomial Root Calculator | Free Online Tool to Solve Roots of 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 The polynomial can be up to fifth degree, so have five zeros at maximum. This website's owner is mathematician Milo Petrovi. To find [latex]f\left(k\right)[/latex], determine the remainder of the polynomial [latex]f\left(x\right)[/latex] when it is divided by [latex]x-k[/latex]. Since polynomial with real coefficients. 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. Please enter one to five zeros separated by space. Since [latex]x-{c}_{\text{1}}[/latex] is linear, the polynomial quotient will be of degree three. [latex]\begin{array}{l}100=a\left({\left(-2\right)}^{4}+{\left(-2\right)}^{3}-5{\left(-2\right)}^{2}+\left(-2\right)-6\right)\hfill \\ 100=a\left(-20\right)\hfill \\ -5=a\hfill \end{array}[/latex], [latex]f\left(x\right)=-5\left({x}^{4}+{x}^{3}-5{x}^{2}+x - 6\right)[/latex], [latex]f\left(x\right)=-5{x}^{4}-5{x}^{3}+25{x}^{2}-5x+30[/latex]. To solve a cubic equation, the best strategy is to guess one of three roots. Calculating the degree of a polynomial with symbolic coefficients. If you're struggling with a math problem, scanning it for key information can help you solve it more quickly. It has two real roots and two complex roots It will display the results in a new window. 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]. 2. Find more Mathematics widgets in Wolfram|Alpha. This pair of implications is the Factor Theorem. Use this calculator to solve polynomial equations with an order of 3 such as ax 3 + bx 2 + cx + d = 0 for x including complex solutions.. Example: with the zeros -2 0 3 4 5, the simplest polynomial is x5-10x4+23x3+34x2-120x. Step 4: If you are given a point that. Create the term of the simplest polynomial from the given zeros. Like any constant zero can be considered as a constant polynimial. For us, the most interesting ones are: If the remainder is 0, the candidate is a zero. The best way to do great work is to find something that you're passionate about. Taylor Series Calculator | Instant Solutions - Voovers [latex]\begin{array}{l}f\left(-x\right)=-{\left(-x\right)}^{4}-3{\left(-x\right)}^{3}+6{\left(-x\right)}^{2}-4\left(-x\right)-12\hfill \\ f\left(-x\right)=-{x}^{4}+3{x}^{3}+6{x}^{2}+4x - 12\hfill \end{array}[/latex]. 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: http://cnx.org/contents/9b08c294-057f-4201-9f48-5d6ad992740d@5.2. Use the Rational Zero Theorem to find the rational zeros of [latex]f\left(x\right)=2{x}^{3}+{x}^{2}-4x+1[/latex]. How to find all the roots (or zeros) of a polynomial checking my quartic equation answer is correct. Find a fourth degree polynomial with real coefficients that has zeros of 3, 2, i, such that [latex]f\left(-2\right)=100[/latex]. 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. Log InorSign Up. Find a Polynomial Given its Graph Questions with Solutions Just enter the expression in the input field and click on the calculate button to get the degree value along with show work. We can now find the equation using the general cubic function, y = ax3 + bx2 + cx+ d, and determining the values of a, b, c, and d. A polynomial equation is an equation formed with variables, exponents and coefficients. Similarly, two of the factors from the leading coefficient, 20, are the two denominators from the original rational roots: 5 and 4. Step 2: Click the blue arrow to submit and see the result! can be used at the function graphs plotter. Lists: Family of sin Curves. 1. It can be written as: f (x) = a 4 x 4 + a 3 x 3 + a 2 x 2 +a 1 x + a 0. The Fundamental Theorem of Algebra states that there is at least one complex solution, call it [latex]{c}_{1}[/latex]. To answer this question, I have to remember that the polynomial's degree gives me the ceiling on the number of bumps. About Press Copyright Contact us Creators Advertise Developers Terms Privacy Policy & Safety How YouTube works Test new features Press Copyright Contact us Creators . In the notation x^n, the polynomial e.g. Continue to apply the Fundamental Theorem of Algebra until all of the zeros are found. (x + 2) = 0. The sheet cake pan should have dimensions 13 inches by 9 inches by 3 inches. Find the polynomial of least degree containing all of the factors found in the previous step. The calculator generates polynomial with given roots. 2. powered by. 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). We can determine which of the possible zeros are actual zeros by substituting these values for xin [latex]f\left(x\right)[/latex]. Zeros Calculator Solved Find a fourth degree polynomial function f(x) with | Chegg.com = x 2 - (sum of zeros) x + Product of zeros. Dividing by [latex]\left(x+3\right)[/latex] gives a remainder of 0, so 3 is a zero of the function. Recall that the Division Algorithm tells us [latex]f\left(x\right)=\left(x-k\right)q\left(x\right)+r[/latex]. 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. If iis a zero of a polynomial with real coefficients, then imust also be a zero of the polynomial because iis the complex conjugate of i. Generate polynomial from roots calculator - Mathportal.org Zero, one or two inflection points. [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]. Please tell me how can I make this better. For fto have real coefficients, [latex]x-\left(a-bi\right)[/latex]must also be a factor of [latex]f\left(x\right)[/latex]. = x 2 - 2x - 15. 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. You can get arithmetic support online by visiting websites such as Khan Academy or by downloading apps such as Photomath. We will be discussing how to Find the fourth degree polynomial function with zeros calculator in this blog post. 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]. [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]. Step 3: If any zeros have a multiplicity other than 1, set the exponent of the matching factor to the given multiplicity. Use the Remainder Theorem to evaluate [latex]f\left(x\right)=2{x}^{5}+4{x}^{4}-3{x}^{3}+8{x}^{2}+7[/latex] Dividing by [latex]\left(x - 1\right)[/latex]gives a remainder of 0, so 1 is a zero of the function. This calculator allows to calculate roots of any polynom of the fourth degree. Determine which possible zeros are actual zeros by evaluating each case of [latex]f\left(\frac{p}{q}\right)[/latex]. If you're struggling with your homework, our Homework Help Solutions can help you get back on track. Example 03: Solve equation $ 2x^2 - 10 = 0 $. These are the possible rational zeros for the function. This free math tool finds the roots (zeros) of a given polynomial. Mathematical problems can be difficult to understand, but with a little explanation they can be easy to solve. The factors of 4 are: Divisors of 4: +1, -1, +2, -2, +4, -4 So the possible polynomial roots or zeros are 1, 2 and 4. Solve real-world applications of polynomial equations. 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 have now introduced a variety of tools for solving polynomial equations. Therefore, [latex]f\left(2\right)=25[/latex]. Also note the presence of the two turning points. 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.

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