find the fourth degree polynomial with zeros calculator find the fourth degree polynomial with zeros calculator

[latex]f\left(x\right)[/latex]can be written as [latex]\left(x - 1\right){\left(2x+1\right)}^{2}[/latex]. Lists: Plotting a List of Points. [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]. of.the.function). 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. To solve the math question, you will need to first figure out what the question is asking. If 2 + 3iwere given as a zero of a polynomial with real coefficients, would 2 3ialso need to be a zero? 3.6 Zeros of Polynomial Functions - Precalculus 2e - OpenStax 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. We will use synthetic division to evaluate each possible zero until we find one that gives a remainder of 0. The polynomial generator generates a polynomial from the roots introduced in the Roots field. 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). To obtain the degree of a polynomial defined by the following expression : a x 2 + b x + c enter degree ( a x 2 + b x + c) after calculation, result 2 is returned. Taylor Series Calculator | Instant Solutions - Voovers (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! [latex]f\left(x\right)=-\frac{1}{2}{x}^{3}+\frac{5}{2}{x}^{2}-2x+10[/latex]. The polynomial generator generates a polynomial from the roots introduced in the Roots field. 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. Edit: Thank you for patching the camera. If you're struggling to clear up a math equation, try breaking it down into smaller, more manageable pieces. The polynomial can be written as [latex]\left(x - 1\right)\left(4{x}^{2}+4x+1\right)[/latex]. Use any other point on the graph (the y -intercept may be easiest) to determine the stretch factor. This process assumes that all the zeroes are real numbers. Solving math equations can be tricky, but with a little practice, anyone can do it! Let the polynomial be ax 2 + bx + c and its zeros be and . Welcome to MathPortal. 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]. I love spending time with my family and friends. 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. Find the fourth degree polynomial with zeros calculator Hence complex conjugate of i is also a root. It has two real roots and two complex roots It will display the results in a new window. Question: Find the fourth-degree polynomial function with zeros 4, -4 , 4i , and -4i. Use the Linear Factorization Theorem to find polynomials with given zeros. 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. Solving Quartic, or 4th Degree, Equations - Study.com Solving matrix characteristic equation for Principal Component Analysis. Math can be tough to wrap your head around, but with a little practice, it can be a breeze! 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]. It's an amazing app! Recall that the Division Algorithm tells us [latex]f\left(x\right)=\left(x-k\right)q\left(x\right)+r[/latex]. Function zeros calculator. We can confirm the numbers of positive and negative real roots by examining a graph of the function. This is called the Complex Conjugate Theorem. It tells us how the zeros of a polynomial are related to the factors. 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. Find the polynomial with integer coefficients having zeroes $ 0, \frac{5}{3}$ and $-\frac{1}{4}$. We can check our answer by evaluating [latex]f\left(2\right)[/latex]. The 4th Degree Equation Calculator, also known as a Quartic Equation Calculator allows you to calculate the roots of a fourth-degree equation. Polynomial Regression Calculator Function's variable: Examples. = x 2 - (sum of zeros) x + Product of zeros. We can provide expert homework writing help on any subject. I really need help with this problem. 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]. Since 1 is not a solution, we will check [latex]x=3[/latex]. Reference: By the fundamental Theorem of Algebra, any polynomial of degree 4 can be written in the form: P(x) = A(x-alpha)(x-beta)(x-gamma) (x-delta) Where, alpha,beta,gamma,delta are the roots (or zeros) of the equation P(x)=0 We are given that -sqrt(11) and 2i are solutions (presumably, although not explicitly stated, of P(x)=0, thus, wlog, we . This calculator allows to calculate roots of any polynom of the fourth degree. powered by "x" x "y" y "a . Find a Polynomial Function Given the Zeros and. 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. Did not begin to use formulas Ferrari - not interestingly. Solving the equations is easiest done by synthetic division. Get the best Homework answers from top Homework helpers in the field. Finding roots of a polynomial equation p(x) = 0; Finding zeroes of a polynomial function p(x) Factoring a polynomial function p(x) There's a factor for every root, and vice versa. Which polynomial has a double zero of $5$ and has $\frac{2}{3}$ as a simple zero? Coefficients can be both real and complex numbers. How do you find the domain for the composition of two functions, How do you find the equation of a circle given 3 points, How to find square root of a number by prime factorization method, Quotient and remainder calculator with exponents, Step functions common core algebra 1 homework, Unit 11 volume and surface area homework 1 answers. 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]. It is used in everyday life, from counting to measuring to more complex calculations. We can then set the quadratic equal to 0 and solve to find the other zeros of the function. Find a fourth-degree polynomial with integer coefficients that has zeros 2i and 1, with 1 a zero of multiplicity 2. By taking a step-by-step approach, you can more easily see what's going on and how to solve the problem. This polynomial function has 4 roots (zeros) as it is a 4-degree function. [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]. 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]. Find the remaining factors. 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 remainder is [latex]25[/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 4}}\hfill \end{array}[/latex]. Quartic equations are actually quite common within computational geometry, being used in areas such as computer graphics, optics, design and manufacturing. Calculator Use. 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, the degree is 6, so the highest number of bumps the graph could have would be 6 1 = 5.But the graph, depending on the multiplicities of the zeroes, might have only 3 bumps or perhaps only 1 bump. 2. powered by. Further polynomials with the same zeros can be found by multiplying the simplest polynomial with a factor. [latex]\begin{array}{l}2x+1=0\hfill \\ \text{ }x=-\frac{1}{2}\hfill \end{array}[/latex]. If possible, continue until the quotient is a quadratic. As we can see, a Taylor series may be infinitely long if we choose, but we may also . 4. Find the zeros of the quadratic function. Zero, one or two inflection points. The factors of 1 are [latex]\pm 1[/latex] and the factors of 2 are [latex]\pm 1[/latex] and [latex]\pm 2[/latex]. (Remember we were told the polynomial was of degree 4 and has no imaginary components). 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. x4+. Once you understand what the question is asking, you will be able to solve it. Example 04: Solve the equation $ 2x^3 - 4x^2 - 3x + 6 = 0 $. If you need help, our customer service team is available 24/7. 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. 4. If you're struggling with your homework, our Homework Help Solutions can help you get back on track. 2. [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. Zero, one or two inflection points. Solution The graph has x intercepts at x = 0 and x = 5 / 2. Where: a 4 is a nonzero constant. 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. However, with a little practice, they can be conquered! For the given zero 3i we know that -3i is also a zero since complex roots occur in Find the fourth degree polynomial function with zeros calculator Use synthetic division to divide the polynomial by [latex]x-k[/latex]. The examples are great and work. 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. I haven't met any app with such functionality and no ads and pays. A fourth degree polynomial is an equation of the form: y = ax4 + bx3 +cx2 +dx +e y = a x 4 + b x 3 + c x 2 + d x + e where: y = dependent value a, b, c, and d = coefficients of the polynomial e = constant adder x = independent value Polynomial Calculators Second Degree Polynomial: y = ax 2 + bx + c Third Degree Polynomial : y = ax 3 + bx 2 + cx + d Roots =. Repeat step two using the quotient found from synthetic division. . This is the standard form of a quadratic equation, Example 01: Solve the equation $ 2x^2 + 3x - 14 = 0 $. We can write the polynomial quotient as a product of [latex]x-{c}_{\text{2}}[/latex] and a new polynomial quotient of degree two. The polynomial can be written as [latex]\left(x+3\right)\left(3{x}^{2}+1\right)[/latex]. Show Solution. Solve real-world applications of polynomial equations. 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. So, the end behavior of increasing without bound to the right and decreasing without bound to the left will continue. For the given zero 3i we know that -3i is also a zero since complex roots occur in. 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! 1. This website's owner is mathematician Milo Petrovi. Now we apply the Fundamental Theorem of Algebra to the third-degree polynomial quotient. 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. It will have at least one complex zero, call it [latex]{c}_{\text{2}}[/latex]. Fourth Degree 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. We name polynomials according to their degree. INSTRUCTIONS: I tried to find the way to get the equation but so far all of them require a calculator. We can now use polynomial division to evaluate polynomials using the Remainder Theorem. Therefore, [latex]f\left(x\right)[/latex] has nroots if we allow for multiplicities. 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. The calculator computes exact solutions for quadratic, cubic, and quartic equations. We can use synthetic division to show that [latex]\left(x+2\right)[/latex] is a factor of the polynomial. the degree of polynomial $ p(x) = 8x^\color{red}{2} + 3x -1 $ is $\color{red}{2}$. Lets use these tools to solve the bakery problem from the beginning of the section. We have now introduced a variety of tools for solving polynomial equations. Really good app for parents, students and teachers to use to check their math work. Note that [latex]\frac{2}{2}=1[/latex]and [latex]\frac{4}{2}=2[/latex], which have already been listed, so we can shorten our list. In this case, a = 3 and b = -1 which gives . You can get arithmetic support online by visiting websites such as Khan Academy or by downloading apps such as Photomath. Since polynomial with real coefficients. Dividing by [latex]\left(x+3\right)[/latex] gives a remainder of 0, so 3 is a zero of the function. Coefficients can be both real and complex numbers. If you need an answer fast, you can always count on Google. Its important to keep them in mind when trying to figure out how to Find the fourth degree polynomial function with zeros calculator. We use cookies to improve your experience on our site and to show you relevant advertising. There are four possibilities, as we can see below. Make Polynomial from Zeros - Rechneronline Now we use $ 2x^2 - 3 $ to find remaining roots. quadratic - degree 2, Cubic - degree 3, and Quartic - degree 4. (I would add 1 or 3 or 5, etc, if I were going from the number . Polynomial Functions of 4th Degree. The factors of 3 are [latex]\pm 1[/latex] and [latex]\pm 3[/latex]. 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]. 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. Are zeros and roots the same? 1, 2 or 3 extrema. Get detailed step-by-step answers This is the first method of factoring 4th degree polynomials. How do you find a fourth-degree polynomial equation, with integer 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. If you want to contact me, probably have some questions, write me using the contact form or email me on = x 2 - 2x - 15. of.the.function). [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]. A "root" (or "zero") is where the polynomial is equal to zero: Put simply: a root is the x-value where the y-value equals zero. of.the.function). 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 can be a difficult subject for some students, but with practice and persistence, anyone can master it. Solve each factor. Continue to apply the Fundamental Theorem of Algebra until all of the zeros are found. This problem can be solved by writing a cubic function and solving a cubic equation for the volume of the cake. 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. Quartic equation Calculator - High accuracy calculation To solve a cubic equation, the best strategy is to guess one of three roots. 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 Free time to spend with your family and friends. Use the Factor Theorem to find the zeros of [latex]f\left(x\right)={x}^{3}+4{x}^{2}-4x - 16[/latex]given that [latex]\left(x - 2\right)[/latex]is a factor of the polynomial. The Fundamental Theorem of Algebra states that there is at least one complex solution, call it [latex]{c}_{1}[/latex]. 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). So for your set of given zeros, write: (x - 2) = 0. into [latex]f\left(x\right)[/latex]. Methods for Finding Zeros of Polynomials | College Algebra - Lumen Learning How To Form A Polynomial With The Given Zeroes - A Plus - A Plus Topper Find the fourth degree polynomial function with zeros calculator For us, the most interesting ones are: quadratic - degree 2, Cubic - degree 3, and Quartic - degree 4. Polynomials: Sums and Products of Roots - mathsisfun.com 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]. Let's sketch a couple of polynomials. Mathematics is a way of dealing with tasks that involves numbers and equations. Polynomial Graphing: Degrees, Turnings, and "Bumps" | Purplemath Input the roots here, separated by comma. No general symmetry. This tells us that kis a zero. 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. Zeros of a polynomial calculator - Polynomial = 3x^2+6x-1 find Zeros of a polynomial, step-by-step online. Use synthetic division to find the zeros of a polynomial function. If you need help, don't hesitate to ask for it. The sheet cake pan should have dimensions 13 inches by 9 inches by 3 inches. Find the zeros of [latex]f\left(x\right)=4{x}^{3}-3x - 1[/latex]. 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). 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] The graph is shown at right using the WINDOW (-5, 5) X (-2, 16). If you're struggling with a math problem, scanning it for key information can help you solve it more quickly. Because [latex]x=i[/latex]is a zero, by the Complex Conjugate Theorem [latex]x=-i[/latex]is also a zero. According to the rule of thumbs: zero refers to a function (such as a polynomial), and the root refers to an equation. Work on the task that is interesting to you. Algebra Polynomial Division Calculator Step 1: Enter the expression you want to divide into the editor. 2. Cubic Equation Calculator Finding 4th Degree Polynomial Given Zeroes - YouTube This theorem forms the foundation for solving polynomial equations. Loading. When the leading coefficient is 1, the possible rational zeros are the factors of the constant term. Zero to 4 roots. Find a basis for the orthogonal complement of w in p2 with the inner product, General solution of differential equation depends on, How do you find vertical asymptotes from an equation, Ovulation calculator average cycle length. Mathematical problems can be difficult to understand, but with a little explanation they can be easy to solve. 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. Write the function in factored form. Can't believe this is free it's worthmoney. 4th Degree Polynomials Division Calculation - MYMATHTABLES.COM b) This polynomial is partly factored. 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. Lets walk through the proof of the theorem. 4th Degree Equation Solver Here is the online 4th degree equation solver for you to find the roots of the fourth-degree equations. We already know that 1 is a zero. Search our database of more than 200 calculators. Log InorSign Up. These zeros have factors associated with them. The Factor Theorem is another theorem that helps us analyze polynomial equations. Find a fourth Find a fourth-degree polynomial function with zeros 1, -1, i, -i. The volume of a rectangular solid is given by [latex]V=lwh[/latex]. This allows for immediate feedback and clarification if needed. If you need your order fast, we can deliver it to you in record time. Finding the x -Intercepts of a Polynomial Function Using a Graph Find the x -intercepts of h(x) = x3 + 4x2 + x 6. Just enter the expression in the input field and click on the calculate button to get the degree value along with show work. Enter the equation in the fourth degree equation. Synthetic division gives a remainder of 0, so 9 is a solution to the equation. Notice that a cubic polynomial has four terms, and the most common factoring method for such polynomials is factoring by grouping. Use synthetic division to check [latex]x=1[/latex]. Find a polynomial that has zeros $ 4, -2 $. An 4th degree polynominals divide calcalution. Quartic Function / Curve: Definition, Examples - Statistics How To The zeros of [latex]f\left(x\right)[/latex]are 3 and [latex]\pm \frac{i\sqrt{3}}{3}[/latex]. Use the Rational Zero Theorem to find the rational zeros of [latex]f\left(x\right)={x}^{3}-3{x}^{2}-6x+8[/latex]. There are two sign changes, so there are either 2 or 0 positive real roots. Polynomial Roots Calculator that shows work - MathPortal The solver will provide step-by-step instructions on how to Find the fourth degree polynomial function with zeros calculator. [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]. The remainder is the value [latex]f\left(k\right)[/latex]. 3. The number of positive 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. 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. Hence the polynomial formed. 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]. To solve a math equation, you need to decide what operation to perform on each side of the equation. 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: Use the factors to determine the zeros of the polynomial. The 4th Degree Equation calculator Is an online math calculator developed by calculator to support with the development of your mathematical knowledge. Polynomial Degree Calculator - Symbolab We will be discussing how to Find the fourth degree polynomial function with zeros calculator in this blog post. Using factoring we can reduce an original equation to two simple equations. The calculator generates polynomial with given roots. The best way to download full math explanation, it's download answer here. Use the zeros to construct the linear factors of the polynomial. Use the Rational Zero Theorem to find the rational zeros of [latex]f\left(x\right)=2{x}^{3}+{x}^{2}-4x+1[/latex]. Only multiplication with conjugate pairs will eliminate the imaginary parts and result in real coefficients. Sol. 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].