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# how to find the zeros of a function equation

Example 1. Find an equation in the form of f(x) = ax^2 + bx + c b. But instead of doing it that way, we might take this as a clue that maybe we can factor by grouping. as a difference of squares if you view two as a This means . Substitute $x=-2$ and $f\left(2\right)=100$ to be equal to zero. A vital implication of the Fundamental Theorem of AlgebraÂ is that a polynomial function of degree nÂ will have nÂ zeros in the set of complex numbers if we allow for multiplicities. Let me just write equals. If the polynomial is written in descending order, Descartesâ Rule of Signs tells us of a relationship between the number of sign changes in $f\left(x\right)$ and the number of positive real zeros. Find the zeros of an equation using this calculator. an x-squared plus nine. After we've factored out an x, we have two second-degree terms. $\begin{array}{lll}f\left(x\right) & =6{x}^{4}-{x}^{3}-15{x}^{2}+2x - 7 \\ f\left(2\right) & =6{\left(2\right)}^{4}-{\left(2\right)}^{3}-15{\left(2\right)}^{2}+2\left(2\right)-7 \\ f\left(2\right) & =25\hfill \end{array}$. Recall that the Division Algorithm states that given a polynomial dividend f(x)Â and a non-zero polynomial divisor d(x)Â where the degree ofÂ d(x) is less than or equal to the degree of f(x), there exist unique polynomials q(x)Â and r(x)Â such that, $f\left(x\right)=d\left(x\right)q\left(x\right)+r\left(x\right)$, If the divisor, d(x), is x âÂ k, this takes the form, $f\left(x\right)=\left(x-k\right)q\left(x\right)+r$, Since the divisor x âÂ kÂ is linear, the remainder will be a constant, r. And, if we evaluate this for x =Â k, we have, $\begin{array}{l}f\left(k\right)=\left(k-k\right)q\left(k\right)+r\hfill \\ \text{}f\left(k\right)=0\cdot q\left(k\right)+r\hfill \\ \text{}f\left(k\right)=r\hfill \end{array}$. So I like to factor that Find -a/2b and f(-a/2b). Find the zeros of an equation … The Fundamental Theorem of Algebra states that, if $f(x)$ is a polynomial of degree $n>0$, then $f(x)$ has at least one complex zero. Then close the parentheses. Since $x-{c}_{\text{1}}$ is linear, the polynomial quotient will be of degree three. This video demonstrates how to find the zeros of a function using any of the TI-84 Series graphing calculators. Now, can x plus the square of two to both sides, you get x is equal to The factors of 1 are $\pm 1$ and the factors of 2 are $\pm 1$ and $\pm 2$. The Factor Theorem is another theorem that helps us analyze polynomial equations. Use the Rational Zero Theorem to find the rational zeros of $f\left(x\right)={x}^{3}-3{x}^{2}-6x+8$. This tells us that kÂ is a zero. If the remainder is 0, the candidate is a zero. and see if you can reverse the distributive property twice. The bakery wants the volume of a small cake to be 351 cubic inches. This one is completely These correspond to the points where the graph crosses the x-axis. $\begin{array}{l}f\left(-x\right)=-{\left(-x\right)}^{4}-3{\left(-x\right)}^{3}+6{\left(-x\right)}^{2}-4\left(-x\right)-12\hfill \\ f\left(-x\right)=-{x}^{4}+3{x}^{3}+6{x}^{2}+4x - 12\hfill \end{array}$. I'm just recognizing this −1+⋯+. Since 1 is not a solution, we will check $x=3$. The zero of a linear function can be found by replacing the y with zero and then solving for x. Maybe my algorithm is not suficient to find roots af any function in any condition, but sampling is an analytical task one must do in every case, not only to find roots. thing to think about. This is an algebraic way to find the zeros of the function f(x). Letâs begin with â3. Here, " ax² + bx +c = 0" is called as quadratic equation. nine from both sides, you get x-squared is So either the multiplicity of $x=-3$ is 1 and there are two complex solutions, which is what we found, or the multiplicity at $x=-3$ is three. that make the polynomial equal to zero. two is equal to zero. The factors of â1 are $\pm 1$Â and the factors of 4 are $\pm 1,\pm 2$, and $\pm 4$. this a little bit simpler. Divide each term by and simplify. This means that we can factor the polynomial function into nÂ factors. Thus, all the x-intercepts for the function are shown. The quadratic is a perfect square. So those are my axes. Dividing by $\left(x - 1\right)$Â gives a remainder of 0, so 1 is a zero of the function. But, if it has some imaginary zeros, it won't have five real zeros. Determine all possible values of $\frac{p}{q}$, where. In this section, we will discuss a variety of tools for writing polynomial functions and solving polynomial equations. The possible values for $\frac{p}{q}$ are $\pm 1$ and $\pm \frac{1}{2}$. Since 3 is not a solution either, we will test $x=9$. The zero of the function is where the y-value is zero. 1 +0. There are some imaginary to be the three times that we intercept the x-axis. 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. So, we can rewrite this as x times x to the fourth power plus nine x-squared minus two x-squared minus 18 is equal to zero. arbitrary polynomial here. $\begin{array}{l}\frac{p}{q}=\frac{\text{Factors of the constant term}}{\text{Factor of the leading coefficient}}\hfill \\ \text{}\frac{p}{q}=\frac{\text{Factors of 3}}{\text{Factors of 3}}\hfill \end{array}$. Look at the graph of the function f. Notice, at $x=-0.5$, the graph bounces off the x-axis, indicating the even multiplicity (2,4,6â¦) for the zero â0.5.Â At $x=1$, the graph crosses the x-axis, indicating the odd multiplicity (1,3,5â¦) for the zero $x=1$. I factor out an x-squared, I'm gonna get an x-squared plus nine. plus nine equal zero? Two possible methods for solving quadratics are factoring and using the quadratic formula. $f\left(x\right)=-\frac{1}{2}{x}^{3}+\frac{5}{2}{x}^{2}-2x+10$. The Rational Zero Theorem helps us to narrow down the list of possible rational zeros for a polynomial function. The constant term is â4; the factors of â4 are $p=\pm 1,\pm 2,\pm 4$. And group together these second two terms and factor something interesting out? The volume of a rectangular solid is given by $V=lwh$. The zeros of a function f are found by solving the equation f(x) = 0. While it might not be as straightforward as solving a quadratic equation, there are a couple of methods you can use to find the solution to a cubic equation without resorting to pages and pages of detailed algebra. Linear functions will have at most one zero. It's gonna be x-squared, if So, there we have it. Anyway, thank you a … that makes the function equal to zero. The number of positive real zeros is either equal to the number of sign changes of $f\left(x\right)$ or is less than the number of sign changes by an even integer. might jump out at you is that all of these Let fÂ be a polynomial function with real coefficients and suppose $a+bi\text{, }b\ne 0$,Â is a zero of $f\left(x\right)$.Â Then, by the Factor Theorem, $x-\left(a+bi\right)$Â is a factor of $f\left(x\right)$.Â For fÂ to have real coefficients, $x-\left(a-bi\right)$Â must also be a factor of $f\left(x\right)$.Â This is true because any factor other than $x-\left(a-bi\right)$,Â when multiplied by $x-\left(a+bi\right)$,Â will leave imaginary components in the product. $\begin{array}{l}\\ 2\overline{)\begin{array}{lllllllll}6\hfill & -1\hfill & -15\hfill & 2\hfill & -7\hfill \\ \hfill & \text{ }12\hfill & \text{ }\text{ }\text{ }22\hfill & 14\hfill & \text{ }\text{ }32\hfill \end{array}}\\ \begin{array}{llllll}\hfill & \text{}6\hfill & 11\hfill & \text{ }\text{ }\text{ }7\hfill & \text{ }\text{ }16\hfill & \text{ }\text{ }25\hfill \end{array}\end{array}$. All right. So, those are our zeros. because this is telling us maybe we can factor out some arbitrary p of x. Find the zeros of $f\left(x\right)=2{x}^{3}+5{x}^{2}-11x+4$. Solving the equations is easiest done by synthetic division. Because $x=i$Â is a zero, by the Complex Conjugate Theorem $x=-i$Â is also a zero. In other words, if a polynomial function fÂ with real coefficients has a complex zero $a+bi$,Â then the complex conjugate $a-bi$Â must also be a zero of $f\left(x\right)$. Real numbers are also complex numbers. List all possible rational zeros of $f\left(x\right)=2{x}^{4}-5{x}^{3}+{x}^{2}-4$. What is the best way to do it? So the function is going X could be equal to zero. these first two terms and factor something interesting out? The zeros are $\text{-4, }\frac{1}{2},\text{ and 1}\text{.}$. Actually, I can even get rid $l=w+4=9+4=13\text{ and }h=\frac{1}{3}w=\frac{1}{3}\left(9\right)=3$. Well, that's going to be a point at which we are intercepting the x-axis. How do I know that? It tells us how the zeros of a polynomial are related to the factors. Find zeros of a polynomial function. there's also going to be imaginary roots, or Use synthetic division to divide the polynomial by $\left(x-k\right)$. - [Voiceover] So, we have a Allowing for multiplicities, a polynomial function will have the same number of factors as its degree. I need to retrieve all the zeros of this function. Letâs walk through the proof of the theorem. just add these two together, and actually that it would be I can factor out an x-squared. If kÂ is a zero, then the remainder rÂ is $f\left(k\right)=0$Â and $f\left(x\right)=\left(x-k\right)q\left(x\right)+0$Â or $f\left(x\right)=\left(x-k\right)q\left(x\right)$. In your textbook, a quadratic function is full of x's and y's.This article focuses on the practical applications of quadratic functions. something out after that. According to Descartesâ Rule of Signs, if we let $f\left(x\right)={a}_{n}{x}^{n}+{a}_{n - 1}{x}^{n - 1}+…+{a}_{1}x+{a}_{0}$Â be a polynomial function with real coefficients: Use Descartesâ Rule of Signs to determine the possible numbers of positive and negative real zeros for $f\left(x\right)=-{x}^{4}-3{x}^{3}+6{x}^{2}-4x - 12$. Yeah, this part right over here and you could add those two middle terms, and then factor in a non-grouping way, and I encourage you to do that. This is a graph of y is equal, y is equal to p of x. Their graphs are always lines. To find the remainder using the Remainder Theorem, use synthetic division to divide the polynomial by $x - 2$. Every equation in the unknown may be rewritten as =by regrouping all the terms in the left-hand side. Write the primary function to accept the coefficients of the polynomial like the C vector above. We have now introduced a variety of tools for solving polynomial equations. Show that $\left(x+2\right)$Â is a factor of ${x}^{3}-6{x}^{2}-x+30$. The possible values for $\frac{p}{q}$, and therefore the possible rational zeros for the function, are $\pm 3, \pm 1, \text{and} \pm \frac{1}{3}$. As you'll learn in the future, The graph of a quadratic function is a parabola. Then we want to think Letâs begin with 1. And let's sort of remind At this x-value the Follow these steps to learn several different ways how to find the zeros of a function. Once we have done this, we can use synthetic division repeatedly to determine all of the zeros of a polynomial function. Khan Academy is a 501(c)(3) nonprofit organization. In the last section, we learned how to divide polynomials. $\begin{array}{l}f\left(x\right)=a\left(x+3\right)\left(x - 2\right)\left(x-i\right)\left(x+i\right)\\ f\left(x\right)=a\left({x}^{2}+x - 6\right)\left({x}^{2}+1\right)\\ f\left(x\right)=a\left({x}^{4}+{x}^{3}-5{x}^{2}+x - 6\right)\end{array}$. It is an equation for the parabola shown higher up. Sure, you add square root Follow these directions to find the intercepts and the zero. So, let me delete that. The zero of a function is a value of x that makes the value of function equal to zero. So, let me give myself We can conclude if kÂ is a zero of $f\left(x\right)$, then $x-k$ is a factor of $f\left(x\right)$. So that's going to be a root. So, the x-values that satisfy this are going to be the roots, or the zeros, and we want the real ones. times x-squared minus two. 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. on the graph of the function, that p of x is going to be equal to zero. To find the other zero, we can set the factor equal to 0. fifth-degree polynomial here, p of x, and we're asked We can determine which of the possible zeros are actual zeros by substituting these values for xÂ in $f\left(x\right)$. We explain Finding the Zeros of a Rational Function with video tutorials and quizzes, using our Many Ways(TM) approach from multiple teachers. A value of x that makes the equation equal to 0 is termed as zeros. 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. Synthetic division can be used to find the zeros of a polynomial function. Also note the presence of the two turning points. You can use your TI-84 Plus calculator to find the zeroes of a function. number of real zeros we have. It will have at least one complex zero, call it ${c}_{\text{2}}$. A value of x that makes the equation equal to 0 is termed as zeros. Rational zeros are also called rational roots and x-intercepts, and are the places on a graph where the function touches the x-axis and has a zero value for the y-axis. any one of them equals zero then I'm gonna get zero. Their zeros are at zero, The zeros of $f\left(x\right)$Â are â3 and $\pm \frac{i\sqrt{3}}{3}$. It actually just jumped out of me as I was writing this down is that we have two third-degree terms. This is just one example problem to show solving quadratic equations by factoring. Letâs begin by multiplying these factors. This is called the Complex Conjugate Theorem. no real solution to this. Notice that two of the factors of the constant term, 6, are the two numerators from the original rational roots: 2 and 3. The zeros of the function are 1 and $-\frac{1}{2}$ with multiplicity 2. By the Factor Theorem, the zeros of ${x}^{3}-6{x}^{2}-x+30$ are â2, 3, and 5. zeros, or there might be. $\begin{array}{l}100=a\left({\left(-2\right)}^{4}+{\left(-2\right)}^{3}-5{\left(-2\right)}^{2}+\left(-2\right)-6\right)\hfill \\ 100=a\left(-20\right)\hfill \\ -5=a\hfill \end{array}$, $f\left(x\right)=-5\left({x}^{4}+{x}^{3}-5{x}^{2}+x - 6\right)$, $f\left(x\right)=-5{x}^{4}-5{x}^{3}+25{x}^{2}-5x+30$. Letâs begin by testing values that make the most sense as dimensions for a small sheet cake. So root is the same thing as a zero, and they're the x-values Therefore, $f\left(x\right)$ has nÂ roots if we allow for multiplicities. And that's why I said, there's Divide by . Polynomialsare functions of general form ( )= +. When x is equal to zero, this If the polynomial is divided by x âÂ k, the remainder may be found quickly by evaluating the polynomial function at k, that is, f(k). We can write the polynomial quotient as a product of $x-{c}_{\text{2}}$ and a new polynomial quotient of degree two. $\begin{array}{l}\frac{p}{q}=\frac{\text{Factors of the constant term}}{\text{Factors of the leading coefficient}}\hfill \\ \text{}\frac{p}{q}=\frac{\text{Factors of -1}}{\text{Factors of 4}}\hfill \end{array}$. So, let's see if we can do that. How To: Given a polynomial function f f, use synthetic division to find its zeros Use the Rational Zero Theorem to list all possible rational zeros of the function. Similarly, two of the factors from the leading coefficient, 20, are the two denominators from the original rational roots: 5 and 4. The number of negative real zeros is either equal to the number of sign changes of $f\left(-x\right)$ or is less than the number of sign changes by an even integer. A parabola can cross the x-axis once, twice, or never.These points of intersection are called x-intercepts or zeros. Well, if you subtract and we'll figure it out for this particular polynomial. X plus the square root of two equal zero. If you want more example please click the below link. If you're seeing this message, it means we're having trouble loading external resources on our website. Clearly f(x) is a quadratic function. Substitute the given volume into this equation. All three of these concepts can be seen by looking at a linear graph. $\begin{array}{l}V=\left(w+4\right)\left(w\right)\left(\frac{1}{3}w\right)\\ V=\frac{1}{3}{w}^{3}+\frac{4}{3}{w}^{2}\end{array}$. Find the Zeros of an Equation with a TI-84 Calculator -- TI84+ SE Tips and Tricks - Duration: 2:32. santybm 63,866 views. $\begin{array}{l}2x+1=0\hfill \\ \text{ }x=-\frac{1}{2}\hfill \end{array}$. Letâs write the volume of the cake in terms of width of the cake. Find the remaining factors. And then they want us to Because y = 0 at these solutions, these zeros (solutions) are really just the x -coordinates of the x -intercepts of the graph of y = f ( x ). As we'll see, it's Now we apply the Fundamental Theorem of Algebra to the third-degree polynomial quotient. Solution to Example 1 To find the zeros of function f, solve the equation f(x) = -2x + 4 = 0 Hence the zero of f is give by x = 2 Example 2 Find the zeros of the quadratic function f is given by This theorem forms the foundation for solving polynomial equations. Linear Functions are functions that can be put into the form y=mx+b. Use the zeros to construct the linear factors of the polynomial. If you are trying to find the zeros for the function (that is find x when f(x) = 0), then that is simply done using quadratic equation - no need for math software. The Rational Zero Theorem tells us that if $\frac{p}{q}$ is a zero of $f\left(x\right)$, then pÂ is a factor of â1 andÂ qÂ is a factor of 4. There will be four of them and each one will yield a factor of $f\left(x\right)$. terms are divisible by x. There are four possibilities, as we can see below. The Fundamental Theorem of Algebra tells us that every polynomial function has at least one complex zero. And so those are going And then over here, if I factor out a, let's see, negative two. Synthetic division gives a remainder of 0, so 9 is a solution to the equation. For a simple linear function, this is very easy. The expression on the calculator is zeros (expression,var) where “expression” is your function and “var” is the variable you want to find zeros for (i.e. Write the polynomial as the product of $\left(x-k\right)$ and the quadratic quotient. Now, it might be tempting to We can check our answer by evaluating $f\left(2\right)$. P of zero is zero. If you're seeing this message, it means we're having trouble loading external resources on our website. It can also be said as the roots of the polynomial equation. There are two sign changes, so there are either 2 or 0 positive real roots. We have figured out our zeros. Use the Factor Theorem to solve a polynomial equation. Polynomial expressions, equations, & functions. Find the zeros of an equation using this calculator. But just to see that this makes sense that zeros really are the x-intercepts. about how many times, how many times we intercept the x-axis. X-squared plus nine equal zero. $\begin{array}{l}3{x}^{2}+1=0\hfill \\ \text{ }{x}^{2}=-\frac{1}{3}\hfill \\ \text{ }x=\pm \sqrt{-\frac{1}{3}}=\pm \frac{i\sqrt{3}}{3}\hfill \end{array}$. Continue to apply the Fundamental Theorem of Algebra until all of the zeros are found. A quadratic function has 2 zeros of 1 and 1. I agree with you, but I can't provide a general rule of sampling to be sure we will get all of the roots. So far we've been able to factor it as x times x-squared plus nine Zeros of a Function on the TI 89 Steps Use the Zeros Function on the TI-89 to find roots (or zeros) easily. Find roots or zeros of a Polynomial in R Programming – polyroot() Function Last Updated: 12-06-2020 polyroot() function in R Language is used to calculate roots of a polynomial equation. The possible values for $\frac{p}{q}$ are $\pm 1,\pm \frac{1}{2}$, and $\pm \frac{1}{4}$. Substitute $\left(c,f\left(c\right)\right)$ into the function to determine the leading coefficient. function is equal zero. gonna be the same number of real roots, or the same We found that both iÂ and âi were zeros, but only one of these zeros needed to be given. or more of those expressions "are equal to zero", The other zero will have a multiplicity of 2 because the factor is squared. this is equal to zero. This is the final equation in the article: f(x) = 0.25x^2 + x + 2. From here we can see that the function has exactly one zero: x = –1. The polynomial can be written as $\left(x+3\right)\left(3{x}^{2}+1\right)$. $f\left(x\right)=a\left(x-{c}_{1}\right)\left(x-{c}_{2}\right)…\left(x-{c}_{n}\right)$. According to the Factor Theorem, kÂ is a zero of $f\left(x\right)$Â if and only if $\left(x-k\right)$Â is a factor of $f\left(x\right)$. $4{x}^{2}-8x+15-\frac{78}{4x+5}$, Another use for the Remainder Theorem is to test whether a rational number is a zero for a given polynomial, but first we need a pool of rational numbers to test. −1 . So when you want to find the roots of a function you have to set the function equal to zero. Therefore, $f\left(2\right)=25$. 1 is the only rational zero of $f\left(x\right)$. So, we can rewrite this as, and of course all of If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. square root of two-squared. I graphed this polynomial and this is what I got. that we can solve this equation. Use synthetic division to find the zeros of a polynomial function. Zeros Calculator The calculator will find zeros (exact and numerical, real and complex) of the linear, quadratic, cubic, quartic, polynomial, rational, irrational, exponential, logarithmic, trigonometric, hyperbolic, and absolute value function on the given interval. Recall that the Division Algorithm tells us $f\left(x\right)=\left(x-k\right)q\left(x\right)+r$. Find zeros of quadratic equation by using formula Notice, written in this form, xÂ âÂ k is a factor of $f\left(x\right)$. The Linear Factorization Theorem tells us that a polynomial function will have the same number of factors as its degree, and each factor will be of the form (xÂ âÂ c) where cÂ is a complex number. Let us see the next concept on "how to find zeros of quadratic polynomial". Use the factors to determine the zeros of the polynomial. into $f\left(x\right)$. The zeros of a quadratic function are nothing but the two values of "x" when f(x) = 0 or ax² + bx +c = 0. Use the Factor Theorem to find the zeros of $f\left(x\right)={x}^{3}+4{x}^{2}-4x - 16$Â given that $\left(x - 2\right)$Â is a factor of the polynomial. We can use synthetic division to show that $\left(x+2\right)$ is a factor of the polynomial. product of those expressions "are going to be zero if one Does every polynomial have at least one imaginary zero? As we will soon see, a polynomial of degree nÂ in the complex number system will have nÂ zeros. of those green parentheses now, if I want to, optimally, make Solve for . $\frac{p}{q}=\frac{\text{Factors of the constant term}}{\text{Factors of the leading coefficient}}=\pm 1,\pm 2,\pm 4,\pm \frac{1}{2}$. a little bit more space. These are the possible rational zeros for the function. ourselves what roots are. Consider a quadratic function with two zeros, $x=\frac{2}{5}$Â and $x=\frac{3}{4}$. $\begin{array}{l}\text{ }f\left(-1\right)=2{\left(-1\right)}^{3}+{\left(-1\right)}^{2}-4\left(-1\right)+1=4\hfill \\ \text{ }f\left(1\right)=2{\left(1\right)}^{3}+{\left(1\right)}^{2}-4\left(1\right)+1=0\hfill \\ \text{ }f\left(-\frac{1}{2}\right)=2{\left(-\frac{1}{2}\right)}^{3}+{\left(-\frac{1}{2}\right)}^{2}-4\left(-\frac{1}{2}\right)+1=3\hfill \\ \text{ }f\left(\frac{1}{2}\right)=2{\left(\frac{1}{2}\right)}^{3}+{\left(\frac{1}{2}\right)}^{2}-4\left(\frac{1}{2}\right)+1=-\frac{1}{2}\hfill \end{array}$. Begin by writing an equation for the volume of the cake. x or y variables). So we want to know how many times we are intercepting the x-axis. the square root of two. Use the Remainder Theorem to evaluate $f\left(x\right)=2{x}^{5}-3{x}^{4}-9{x}^{3}+8{x}^{2}+2$ So, if you don't have five real roots, the next possibility is We already know that 1 is a zero. Only positive numbers make sense as dimensions for a cake, so we need not test any negative values. We can use this theorem to argue that, if $f\left(x\right)$ is a polynomial of degree $n>0$, and aÂ is a non-zero real number, then $f\left(x\right)$ has exactly nÂ linear factors. 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. And so, here you see, If the polynomial function fÂ has real coefficients and a complex zero of the form $a+bi$,Â then the complex conjugate of the zero, $a-bi$,Â is also a zero. Use various methods in order to find all the zeros of polynomial expressions or functions. Answer to: a. Use the Rational Zero Theorem to list all possible rational zeros of the function. Again, there are two sign changes, so there are either 2 or 0 negative real roots. This problem can be solved by writing a cubic function and solving a cubic equation for the volume of the cake. What should the dimensions of the container be? 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 $\left(x-c\right)$ where cÂ is a complex number. Find the zeros of the quadratic function. 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). And then maybe we can factor 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.