4 14 2,f( 3 2 For us, the most interesting ones are: So we really want to solve x 2 3 3 8x+5 +55 3 x x ( 3 this a little bit simpler. x+6=0, 2 x x The length is 3 inches more than the width. And that is the solution: x = 1/2. Use synthetic division to evaluate a given possible zero by synthetically dividing the candidate into the polynomial. {eq}P(x) = \color{red}a(x-\color{blue}{z_1})(x-\color{blue}{z_2})(x-\color{blue}{z_3}) {/eq}. f(x)=3 x It is known that the product is zero when at least one factor is zero, so we just need to set the factors equal to zero and solve the corresponding equations (some equations have already been solved, some can't be solved by hand). The polynomial generator generates a polynomial from the roots introduced in the Roots field. f(x)= +8 The largest exponent of appearing in is called the degree of . 3 Use the Rational Roots Test to Find All Possible Roots. 2 f(x)= Solve each factor. Adjust the number of factors to match the number of. Check $$$2$$$: divide $$$2 x^{3} + x^{2} - 13 x + 6$$$ by $$$x - 2$$$. Polynomial roots calculator This free math tool finds the roots (zeros) of a given polynomial. 98 Subtract 1 from both sides: 2x = 1. 3 10 f(x)=10 ( +5 ) x Creative Commons Attribution License 2 4 (more notes on editing functions are located below) Direct link to Keerthana Revinipati's post How do you graph polynomi, Posted 5 years ago. x ), Real roots: 2, 2 Standard Form: A form in which the polynomial's terms are arranged from the highest degree to the smallest: {eq}P(x) = ax^n + bx^{n-1} + cx^{n-2} + + yx + z 4 +2 The trailing coefficient (coefficient of the constant term) is $$$-12$$$. x Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License . This is because the exponent on the x is 3, and the exponent on the y is 2. x Find an nth-degree polynomial function with real coefficients - Wyzant Once you've done that, refresh this page to start using Wolfram|Alpha. 2 For the following exercises, use your calculator to graph the polynomial function. 4 P(x) = x^4-15x^3+54x^2+108x-648\\ 3 Step 4a: Remember that we need the whole equation, not just the value of a. Step 2: Click on the "Find" button to find the degree of a polynomial. x 3 - [Voiceover] So, we have a 12 2,10 Use the Linear Factorization Theorem to find polynomials with given zeros. +26x+6. +16 2 x +2 3 2 that make the polynomial equal to zero. 3 x 2 4x+4, f(x)=2 All right. For example, consider g (x)= (x-1)^2 (x-4) g(x) = (x 1)2(x 4). 3 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. \hline f(x)=8 8x+5, f(x)=3 There is a straightforward way to determine the possible numbers of positive and negative real . +3 10x+24=0 FOIL: A process for multiplying two factors with two terms, each. P(x) = \color{red}{(x+3)}\color{blue}{(x-6)}\color{green}{(x-6)}(x-6) & \text{Removing exponents and instead writing out all of our factors can help.} And the whole point 4 x n=3 ; 2 and 5i are zeros; f (1)=-52 Since f (x) has real coefficients 5i is a root, so is -5i So, 2, 5i, and -5i are roots 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). 10x5=0, 4 10x5=0 x 2 )=( x 3 Step 3: If any zeros have a multiplicity other than 1, set the exponent of the matching factor to the given multiplicity. parentheses here for now, If we factor out an x-squared plus nine, it's going to be x-squared plus nine times x-squared, x-squared minus two. +3 For the following exercises, construct a polynomial function of least degree possible using the given information. To subtract polynomials, combine and subtract the coefficients near the like terms: $$$\left(\color{Crimson}{2 x^{4}}\color{BlueViolet}{- 3 x^{3}}\color{GoldenRod}{- 15 x^{2}}+\color{DarkBlue}{32 x}\color{DarkCyan}{-12}\right)-\left(\color{GoldenRod}{x^{2}}\color{DarkBlue}{- 4 x}\color{DarkCyan}{-12}\right)=$$$, $$$=\color{Crimson}{2 x^{4}}\color{BlueViolet}{- 3 x^{3}}+\color{GoldenRod}{\left(\left(-15\right)-1\right) x^{2}}+\color{DarkBlue}{\left(32-\left(-4\right)\right) x}+\color{DarkCyan}{\left(\left(-12\right)-\left(-12\right)\right) }=$$$, $$$=2 x^{4} - 3 x^{3} - 16 x^{2} + 36 x$$$. Not necessarily this p of x, but I'm just drawing The volume is 12 +3 x +57x+85=0, 3 Check $$$2$$$: divide $$$2 x^{4} - 3 x^{3} - 15 x^{2} + 32 x - 12$$$ by $$$x - 2$$$. 9 3 So there's some x-value 2x+8=0, 4 2 an x-squared plus nine. To solve cubic equations, we usually use the factoting method: Example 05: Solve equation $ 2x^3 - 4x^2 - 3x + 6 = 0 $. 2 x 3 3,f( ( 9 An error occurred trying to load this video. x f(x)=10 x + 1 He has worked for nearly 10 years in mathematics education. 5 Get access to thousands of practice questions and explanations! OpenStax is part of Rice University, which is a 501(c)(3) nonprofit. x +1, f(x)=4 2 3 If you're already familiar with multiplying polynomial factors from prior lessons, you may already know how to do this step and can skip down to the end of the table for the standard form. If you want to contact me, probably have some questions, write me using the contact form or email me on + x +11x+10=0 verifying: the point is listed . 2 Check $$$-1$$$: divide $$$2 x^{3} + x^{2} - 13 x + 6$$$ by $$$x + 1$$$. 2 Polynomial Equation Calculator - Symbolab x Please enable JavaScript. + 3x+1=0, 8 f(x)=2 x x Since the remainder is `0`, then $$$2$$$ is the root, and $$$x - 2$$$ is the factor: $$$2 x^{4} - 3 x^{3} - 15 x^{2} + 32 x - 12 = \left(x - 2\right) \left(2 x^{3} + x^{2} - 13 x + 6\right)$$$, $$\color{red}{\left(2 x^{4} - 3 x^{3} - 15 x^{2} + 32 x - 12\right)} = \color{red}{\left(x - 2\right) \left(2 x^{3} + x^{2} - 13 x + 6\right)}$$. +13x+1 x 1, f(x)= The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo 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. If we're on the x-axis x $$$\frac{2 x^{4} - 3 x^{3} - 15 x^{2} + 32 x - 12}{x^{2} - 4 x - 12}=2 x^{2} + 5 x + 29+\frac{208 x + 336}{x^{2} - 4 x - 12}$$$. +13x6;x1, f(x)=2 3 16x80=0, x cubic meters. The quotient is $$$2 x^{3} + x^{2} - 13 x + 6$$$, and the remainder is $$$0$$$ (use the synthetic division calculator to see the steps). f(x)=2 4 2 }\\ 2 x 12x30,2x+5 4x+4 2 x The first one is obvious. 2 At this x-value the So the real roots are the x-values where p of x is equal to zero. 3 x 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: x 25 2 2 x x 16x80=0, x Divide both sides by 2: x = 1/2. In this example, the last number is -6 so our guesses are. x x x x +4 Calculator shows detailed step-by-step explanation on how to solve the problem. x 9 x +25x26=0 x Plus, get practice tests, quizzes, and personalized coaching to help you So, there we have it. f(x)=4 3 +14x5 +26 x 3 Please tell me how can I make this better. x +11. 6 The roots are $$$x_{1} = \frac{1}{2}$$$, $$$x_{2} = -3$$$ (use the quadratic equation calculator to see the steps). The height is greater and the volume is Simplify: $$$2 \left(x - 2\right)^{2} \left(x - \frac{1}{2}\right) \left(x + 3\right)=\left(x - 2\right)^{2} \left(x + 3\right) \left(2 x - 1\right)$$$. x + ax, where the a's are coefficients and x is the variable. \begin{array}{l l l} x x x x 3 11x6=0, 2 (with multiplicity 2) and Now we can split our equation into two, which are much easier to solve. +7 2 If you're seeing this message, it means we're having trouble loading external resources on our website. 3 + x 3 2 For the following exercises, use Descartes Rule to determine the possible number of positive and negative solutions. ( Get unlimited access to over 88,000 lessons. 9 figure out the smallest of those x-intercepts, These are the possible values for `p`. 2 5x+4, f(x)=6 copyright 2003-2023 Study.com. and 2 2,4 Now we use $ 2x^2 - 3 $ to find remaining roots. +13x6;x1 x +8 4 2 +1 2 f(x)=2 When there are multiple terms, such as in a polynomial, we find the degree by looking at each of the terms, getting their individual degrees, then noting the highest one. x 8x+5, f(x)=3 &\text{degree 4 to 3, then to 2, then 1, then 0. x 2 Because our equation now only has two terms, we can apply factoring. 2 This is a topic level video of Finding a Polynomial of a Given Degree with Given Zeros: Real Zeros for ASU.Join us!https://www.edx.org/course/college-algebra. x x However many unique real roots we have, that's however many times we're going to intercept the x-axis. Find a polynomial that has zeros $0, -1, 1, -2, 2, -3$ and $3$. 3 Example: Find the polynomial f (x) of degree 3 with zeros: x = -1, x = 2, x = 4 and f (1) = 8 Show Video Lesson 4 are not subject to the Creative Commons license and may not be reproduced without the prior and express written Step 5: Multiply the factors together using the distributive property to get the standard form. x Same reply as provided on your other question. x Adjust the number of factors to match the number of zeros (write more or erase some as needed). )=( The volume is 120 cubic inches. These use methods from complex analysis as well as sophisticated numerical algorithms, and indeed, this is an area of ongoing research and development. ) 2 2 2 2 5 x This is a graph of y is equal, y is equal to p of x. 6 Find a polynomial of degree 4 with zeros of 1, 7, and -3 (multiplicity 2) and a y-intercept of 4. 3 2 Indeed, if $$$x_1$$$ and $$$x_2$$$ are the roots of the quadratic equation $$$ax^2+bx+c=0$$$, then $$$ax^2+bx+c=a(x-x_1)(x-x_2)$$$. 3,f( 32x15=0 72 (Click on graph to enlarge) f (x) = help (formulas) Find the equation for a polynomial f (x) that satisfies the following: - Degree 3 - Zero at x = 1 - Zero at x = 2 - Zero at x = 2 - y-intercept of (0, 8) f (x) = help (formulas) For the following exercises, find all complex solutions (real and non-real). ( as a difference of squares if you view two as a 2 So the first thing that 1 15x+25. x+6=0, 2 2 There are formulas for . Well, the smallest number here is negative square root, negative square root of two. 2 And let's sort of remind 2 x 3 Our mission is to improve educational access and learning for everyone. x 5 $$$\left(2 x^{4} - 3 x^{3} - 15 x^{2} + 32 x - 12\right)\cdot \left(x^{2} - 4 x - 12\right)=2 x^{6} - 11 x^{5} - 27 x^{4} + 128 x^{3} + 40 x^{2} - 336 x + 144$$$. x 98 In total, I'm lost with that whole ending. Some quadratic factors have no real zeroes, because when solving for the roots, there might be a negative number under the radical. Put this in 2x speed and tell me whether you find it amusing or not. +8x+12=0, x The height is 2 inches greater than the width. The radius is 2 x The volume is 86.625 cubic inches. 3 x Topic: Finding a Polynomial of a Given Degree with Given Zeros: Real x \hline \\ +x+6;x+2 7x+3;x1 f(x)=2 10 4 x The volume is 86.625 cubic inches. x +11 x x x 2 ). x Direct link to Dandy Cheng's post Since it is a 5th degree , Posted 6 years ago. And, if you don't have three real roots, the next possibility is you're And that's why I said, there's The length is twice as long as the width. )=( 1 +x1 x 3 x A "root" is when y is zero: 2x+1 = 0. 3 2 To understand what is meant by multiplicity, take, for example, . +13 +2 x 3 x + 4 x 4x+4, f(x)=2 3 So we want to solve this equation. meter greater than the height. 23x+6 3 2 x Create the term of the simplest polynomial from the given zeros. 4 48 cubic meters. For example, 2 20x+12;x+3 2 ) 4 4 x +26x+6 These are the possible values for `p`. if we plug in $ \color{blue}{x = 2} $ into the equation we get, So, $ \color{blue}{x = 2} $ is the root of the equation. X-squared plus nine equal zero. 2 +22 )=( . + x 4 4 x x x 28.125 3 Non-polynomial functions include trigonometric functions, exponential functions, logarithmic functions, root functions, and more. of those green parentheses now, if I want to, optimally, make So, x could be equal to zero. x }\\ +13x+1 x Then we want to think 24 10x+24=0, 2 +11 +13x+1, f(x)=4 2 \hline \\ 7 3 3 Their zeros are at zero, 4 = a(7)(9) \\ +x+6;x+2 x If k is a zero, then the remainder r is f(k) = 0 and f(x) = (x k)q(x) + 0 or f(x) = (x k)q(x). 2 \end{array} $$. P(x) = \color{#856}{x^3}(x-6)\color{#856}{-9x^2}(x-6)\color{#856}{+108}(x-6) & \text{Next, we distributed the final factor, multiplied it out, and combined like terms, as before. How to Find a Polynomial of a Given Degree with Given Zeros 9 2 can be used at the function graphs plotter. 2 f(x)=16 ) Other operations rely on theorems and algorithms from number theory, abstract algebra and other advanced fields to compute results. f(x)=2 The roots are $$$x_{1} = 6$$$, $$$x_{2} = -2$$$ (use the quadratic equation calculator to see the steps). \hline \\ ), Real roots: 1, 1 (with multiplicity 2 and 1) and 2 Factor it and set each factor to zero. x 3 x 2 x 2 25x+75=0, 2 7 4 3x+1=0 If possible, continue until the quotient is a quadratic. First, find the real roots. ) f(x)= Well, let's just think about an arbitrary polynomial here. +2 Andrew has a master's degree in learning and technology as well as a bachelor's degree in mathematics. +25x26=0, x 3 , 0, ) 3 x Calculator shows detailed step-by-step explanation on how to solve the problem. x3 1 x 3 - 1. x 2 This calculator will allow you compute polynomial roots of any valid polynomial you provide. Two possible methods for solving quadratics are factoring and using the quadratic formula. The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo x x +13x+1, f(x)=4 4 succeed. The volume is 192 cubic inches. If the remainder is not zero, discard the candidate. Example: with the zeros -2 0 3 4 5, the simplest polynomial is x5-10x4+23x3+34x2-120x. For the following exercises, find the dimensions of the box described. 2,f( x Restart your browser. 2 It does it has 3 real roots and 2 imaginary roots. f(x)=3 2 + 2 10 terms are divisible by x. The length is three times the height and the height is one inch less than the width. x 2 Use synthetic division to evaluate a given possible zero by synthetically dividing the candidate into the polynomial. 3 Polynomials Calculator - Symbolab +8x+12=0, x Log in here for access. 2 2 Degree of a Polynomial Calculator | Tool to Find Polynomial Degree Value 3 For equation solving, Wolfram|Alpha calls the Wolfram Language's Solve and Reduce functions, which contain a broad range of methods for all kinds of algebra, from basic linear and quadratic equations to multivariate nonlinear systems. 3 2 As we'll see, it's ) f(x)=2 7 function is equal zero. 10x24=0, x By experience, or simply guesswork. 4 Sure, if we subtract square This is the x-axis, that's my y-axis. To factor the quadratic function $$$2 x^{2} + 5 x - 3$$$, we should solve the corresponding quadratic equation $$$2 x^{2} + 5 x - 3=0$$$. 1 Solve real-world applications of polynomial equations. Remember that a y-intercept has an x-value of 0, so a y-intercept of 4 means the point is (0,4). 3 It also displays the step-by-step solution with a detailed explanation. +20x+8, f(x)=10 The quotient is $$$2 x^{3} - 5 x^{2} - 10 x + 42$$$, and the remainder is $$$-54$$$ (use the synthetic division calculator to see the steps). +16 {/eq} would have a degree of 5. 2 f(x)=5 x +32x+17=0. 32x15=0, 2 x +16 3 2 Notice that a cubic polynomial has four terms, and the most common factoring method for such polynomials is factoring by grouping. 3 In the notation x^n, the polynomial e.g. 20x+12;x+3 +11. For example: {eq}P(x) = (\color{red}a+\color{blue}b)(\color{green}c+\color{purple}d)\\ entering the polynomial into the calculator. 3 +55 2 +8x+12=0 Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals. 8 3 Algebra Examples | Simplifying Polynomials | Finding All Possible Roots +8 For example, if the expression is 5xy+3 then the degree is 1+3 = 4. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License . ) +5 f(x)=2 It also factors polynomials, plots polynomial solution sets and inequalities and more. 12x30,2x+5. +26x+6. 2 4 +2 2 4 ( 16 cubic meters. 3 2 2 4 = a(63) \\ Simplify and remove duplicates (if any): $$$\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12, \pm \frac{1}{2}, \pm \frac{3}{2}$$$. x The radius is 3 inches more than the height. +32x12=0, x +x+1=0 Compute a polynomial from zeros: find polynomial with zeros at 2, 3 determine the polynomial with zeros at 2 and 3 with multiplicities 3 and 4 Expansion Expand polynomial expressions using FOIL and other methods. & \text{Colors are used to improve visibility. To solve a polynomial equation write it in standard form (variables and canstants on one side and zero on the other side of the equation). 2,10 Since the remainder is `0`, then $$$2$$$ is the root, and $$$x - 2$$$ is the factor: $$$2 x^{3} + x^{2} - 13 x + 6 = \left(x - 2\right) \left(2 x^{2} + 5 x - 3\right)$$$, $$\left(x - 2\right) \color{red}{\left(2 x^{3} + x^{2} - 13 x + 6\right)} = \left(x - 2\right) \color{red}{\left(x - 2\right) \left(2 x^{2} + 5 x - 3\right)}$$. I factor out an x-squared, I'm gonna get an x-squared plus nine. 3 3 x 4 14 x x Factorized it is written as (x+2)*x* (x-3)* (x-4)* (x-5). 4 8 It also displays the step-by-step solution with a detailed explanation. ( As you'll learn in the future, 2 2 +2 Thus, we can write that $$$2 x^{4} - 3 x^{3} - 15 x^{2} + 32 x - 12=0$$$ is equivalent to the $$$\left(x - 2\right)^{2} \left(x + 3\right) \left(2 x - 1\right)=0$$$. Why are imaginary square roots equal to zero? P(x) = \color{#856}{(x^3-6x^2-3x^2+18x-18x+108)}(x-6) & \text{FOIL wouldn't have worked here because the first factor has 3 terms. 2 x 4 3 ( 2 Polynomial expressions, equations, & functions. 2 3 x 2 2 2,10 4 If the remainder is 0, the candidate is a zero. +9x9=0 2 gonna be the same number of real roots, or the same 4 Instead, this one has three. 2 3 x The length is three times the height and the height is one inch less than the width. 3 A note: If you are already familiar with the binomial theorem, it can help with multiplying out factors and can be applied in problems like this. 3 2 2 She has abachelors degree in mathematics from the University of Delaware and a Master of Education degree from Wesley College. x It is not saying that the roots = 0. 2,f( Already a subscriber? Therefore, the roots of the initial equation are: $$$x_1=6$$$; $$$x_2=-2$$$. x Online Polynomial Degree Calculator - Cuemath Posted 7 years ago. It actually just jumped out of me as I was writing this down is that we have two third-degree terms. f(x)= +26x+6 Working Backwards from Zeroes to Polynomials - Explained! x And, once again, we just Descartes' Rule of Signs. ), Real roots: 1, 1 (with multiplicity 2 and 1) and factored if we're thinking about real roots. This polynomial is considered to have two roots, both equal to 3. 3 3 x p = 1 p = 1. q = 1 . At this x-value, we see, based If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. 3 5 x about how many times, how many times we intercept the x-axis. +3 two is equal to zero. +2 Generate polynomial from roots calculator - mathportal.org 2 32x15=0 If you are using a graphing utility, use it to graph the function and verify the real zeros and the given function value. x 9;x3 Use the Rational Zero Theorem to find rational zeros. Then close the parentheses. 2 9;x3, x x x As an Amazon Associate we earn from qualifying purchases. Solve the quadratic equation $$$2 x^{2} + 5 x - 3=0$$$. 1 x 2 )=( x $$$\left(\color{DarkCyan}{2 x^{4}}\color{DarkBlue}{- 3 x^{3}}\color{GoldenRod}{- 15 x^{2}}+\color{BlueViolet}{32 x}\color{Crimson}{-12}\right) \cdot \left(\color{DarkMagenta}{x^{2}}\color{OrangeRed}{- 4 x}\color{Chocolate}{-12}\right)=$$$, $$$=\left(\color{DarkCyan}{2 x^{4}}\right)\cdot \left(\color{DarkMagenta}{x^{2}}\right)+\left(\color{DarkCyan}{2 x^{4}}\right)\cdot \left(\color{OrangeRed}{- 4 x}\right)+\left(\color{DarkCyan}{2 x^{4}}\right)\cdot \left(\color{Chocolate}{-12}\right)+$$$, $$$+\left(\color{DarkBlue}{- 3 x^{3}}\right)\cdot \left(\color{DarkMagenta}{x^{2}}\right)+\left(\color{DarkBlue}{- 3 x^{3}}\right)\cdot \left(\color{OrangeRed}{- 4 x}\right)+\left(\color{DarkBlue}{- 3 x^{3}}\right)\cdot \left(\color{Chocolate}{-12}\right)+$$$, $$$+\left(\color{GoldenRod}{- 15 x^{2}}\right)\cdot \left(\color{DarkMagenta}{x^{2}}\right)+\left(\color{GoldenRod}{- 15 x^{2}}\right)\cdot \left(\color{OrangeRed}{- 4 x}\right)+\left(\color{GoldenRod}{- 15 x^{2}}\right)\cdot \left(\color{Chocolate}{-12}\right)+$$$, $$$+\left(\color{BlueViolet}{32 x}\right)\cdot \left(\color{DarkMagenta}{x^{2}}\right)+\left(\color{BlueViolet}{32 x}\right)\cdot \left(\color{OrangeRed}{- 4 x}\right)+\left(\color{BlueViolet}{32 x}\right)\cdot \left(\color{Chocolate}{-12}\right)+$$$, $$$+\left(\color{Crimson}{-12}\right)\cdot \left(\color{DarkMagenta}{x^{2}}\right)+\left(\color{Crimson}{-12}\right)\cdot \left(\color{OrangeRed}{- 4 x}\right)+\left(\color{Crimson}{-12}\right)\cdot \left(\color{Chocolate}{-12}\right)=$$$. x x ourselves what roots are. Finding the Equation of a Polynomial Function - Online Math Learning The number of positive real zeros is either equal to the number of sign changes of, The number of negative real zeros is either equal to the number of sign changes of. 3 )=( + x 3 2 + x 4 2 I, Posted 4 years ago. 4 f(x)=4 3 ). 16x+32, f(x)=2 x \text{Inner = } & \color{blue}b \color{green}c & \text{ because b and c are the terms closest to the middle. The height is one less than one half the radius. Polynomial Generator from Roots - SolveMyMath x x 3 The height is 2 inches greater than the width. )=( The volume is 120 cubic inches. x x Polynomial Roots Calculator This free math tool finds the roots (zeros) of a given polynomial. 4 And you could tackle it the other way. + 25x+75=0 The graph has one zero at x=0, specifically at the point (0, 0). 4 To factor the quadratic function $$$x^{2} - 4 x - 12$$$, we should solve the corresponding quadratic equation $$$x^{2} - 4 x - 12=0$$$. 3 x +57x+85=0, 3 x x +9x9=0, 2 15x+25. x function's equal to zero. 2 x 4 +3 x 48 I designed this website and wrote all the calculators, lessons, and formulas. How to Find a Polynomial of a Given Degree with Given Complex Zeros 4 x +39 +2 2 2 x Based on the graph, find the rational zeros. +37 The volume is }\\ \frac{4}{63} = a{/eq}. x 2 The only way to take the square root of negative numbers is with imaginary numbers, or complex numbers, which results in imaginary roots, or zeroes.