polynomial function in standard form with zeros calculator

Recall that the Division Algorithm. We solved each of these by first factoring the polynomial and then using the zero factor property on the factored form. Enter the given function in the expression tab of the Zeros Calculator to find the zeros of the function. How to: Given a polynomial function \(f(x)\), use the Rational Zero Theorem to find rational zeros. The possible values for \(\frac{p}{q}\) are 1 and \(\frac{1}{2}\). However, #-2# has a multiplicity of #2#, which means that the factor that correlates to a zero of #-2# is represented in the polynomial twice. Similarly, if \(xk\) is a factor of \(f(x)\), then the remainder of the Division Algorithm \(f(x)=(xk)q(x)+r\) is \(0\). 12 Sample Introduction Letters | Format, Examples and How To Write Introduction Letters? This is the essence of the Rational Zero Theorem; it is a means to give us a pool of possible rational zeros. The first one is obvious. Feel free to contact us at your convenience! The monomial is greater if the rightmost nonzero coordinate of the vector obtained by subtracting the exponent tuples of the compared monomials is negative in the case of equal degrees. polynomial in standard form But first we need a pool of rational numbers to test. The degree of a polynomial is the value of the largest exponent in the polynomial. Begin by writing an equation for the volume of the cake. If you're looking for a reliable homework help service, you've come to the right place. Good thing is, it's calculations are really accurate. Here are the steps to find them: Some theorems related to polynomial functions are very helpful in finding their zeros: Here are a few examples of each type of polynomial function: Have questions on basic mathematical concepts? 3. 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). Follow the colors to see how the polynomial is constructed: #"zero at "color(red)(-2)", multiplicity "color(blue)2##"zero at "color(green)4", multiplicity "color(purple)1#, #p(x)=(x-(color(red)(-2)))^color(blue)2(x-color(green)4)^color(purple)1#. Or you can load an example. For a polynomial, if #x=a# is a zero of the function, then # (x-a)# is a factor of the function. A polynomial function is the simplest, most commonly used, and most important mathematical function. Function zeros calculator WebPolynomial Calculator Calculate polynomials step by step The calculator will find (with steps shown) the sum, difference, product, and result of the division of two polynomials (quadratic, binomial, trinomial, etc.). The coefficients of the resulting polynomial can be calculated in the field of rational or real numbers. You are given the following information about the polynomial: zeros. Polynomial For example: 8x5 + 11x3 - 6x5 - 8x2 = 8x5 - 6x5 + 11x3 - 8x2 = 2x5 + 11x3 - 8x2. \[ 2 \begin{array}{|ccccc} \; 6 & 1 & 15 & 2 & 7 \\ \text{} & 12 & 22 & 14 & 32 \\ \hline \end{array} \\ \begin{array}{ccccc} 6 & 11 & \; 7 & \;\;16 & \;\; 25 \end{array} \]. WebA polynomial function in standard form is: f (x) = a n x n + a n-1 x n-1 + + a 2 x 2 + a 1 x + a 0. What are the types of polynomials terms? Write the term with the highest exponent first. Webof a polynomial function in factored form from the zeros, multiplicity, Function Given the Zeros, Multiplicity, and (0,a) (Degree 3). \[ \begin{align*} \dfrac{p}{q}=\dfrac{factor\space of\space constant\space term}{factor\space of\space leading\space coefficient} \\[4pt] &=\dfrac{factor\space of\space 3}{factor\space of\space 3} \end{align*}\]. Lets begin by multiplying these factors. WebFor example: 8x 5 + 11x 3 - 6x 5 - 8x 2 = 8x 5 - 6x 5 + 11x 3 - 8x 2 = 2x 5 + 11x 3 - 8x 2. A quadratic equation has two solutions if the discriminant b^2 - 4ac is positive. The Rational Zero Theorem tells us that if \(\dfrac{p}{q}\) is a zero of \(f(x)\), then \(p\) is a factor of 1 and \(q\) is a factor of 4. Standard Form You don't have to use Standard Form, but it helps. WebTo write polynomials in standard form using this calculator; Enter the equation. These ads use cookies, but not for personalization. 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: This algebraic expression is called a polynomial function in variable x. You are given the following information about the polynomial: zeros. In the event that you need to. Solve each factor. However, #-2# has a multiplicity of #2#, which means that the factor that correlates to a zero of #-2# is represented in the polynomial twice. Write A Polynomial Function In Standard Form With Zeros Calculator | Best Writing Service Degree: Ph.D. Plagiarism report. the possible rational zeros of a polynomial function have the form \(\frac{p}{q}\) where \(p\) is a factor of the constant term and \(q\) is a factor of the leading coefficient. Now we apply the Fundamental Theorem of Algebra to the third-degree polynomial quotient. WebPolynomial Standard Form Calculator - Symbolab New Geometry Polynomial Standard Form Calculator Reorder the polynomial function in standard form step-by-step full pad Lets begin with 3. The Rational Zero Theorem helps us to narrow down the list of possible rational zeros for a polynomial function. Graded lex order examples: Example 02: Solve the equation $ 2x^2 + 3x = 0 $. Write a Polynomial Function from its Zeros Zeros of a polynomial calculator Linear Functions are polynomial functions of degree 1. Polynomial function standard form calculator WebForm a polynomial with given zeros and degree multiplicity calculator. \color{blue}{2x } & \color{blue}{= -3} \\ \color{blue}{x} &\color{blue}{= -\frac{3}{2}} \end{aligned} $$, Example 03: Solve equation $ 2x^2 - 10 = 0 $. Subtract from both sides of the equation. A monomial is is a product of powers of several variables xi with nonnegative integer exponents ai: The factors of 1 are 1 and the factors of 4 are 1,2, and 4. The other zero will have a multiplicity of 2 because the factor is squared. The highest exponent in the polynomial 8x2 - 5x + 6 is 2 and the term with the highest exponent is 8x2. It tells us how the zeros of a polynomial are related to the factors. Function zeros calculator. Zeros Formula: Assume that P (x) = 9x + 15 is a linear polynomial with one variable. Input the roots here, separated by comma. Practice your math skills and learn step by step with our math solver. Find the exponent. A polynomial with zeros x=-6,2,5 is x^3-x^2-32x+60=0 in standard form. Learn the why behind math with our certified experts, Each exponent of variable in polynomial function should be a. Zeros Calculator If you're looking for something to do, why not try getting some tasks? You can choose output variables representation to the symbolic form, indexed variables form, or the tuple of exponents. The highest degree is 6, so that goes first, then 3, 2 and then the constant last: x 6 + 4x 3 + 3x 2 7. Lets begin by testing values that make the most sense as dimensions for a small sheet cake. Double-check your equation in the displayed area. WebZero: A zero of a polynomial is an x-value for which the polynomial equals zero. Example 1: A polynomial function of degree 5 has zeros of 2, -5, 1 and 3-4i.What is the missing zero? 2 x 2x 2 x; ( 3) a = b 10 n.. We said that the number b should be between 1 and 10.This means that, for example, 1.36 10 or 9.81 10 are in standard form, but 13.1 10 isn't because 13.1 is bigger Example: Put this in Standard Form: 3x 2 7 + 4x 3 + x 6. David Cox, John Little, Donal OShea Ideals, Varieties, and Quadratic Equation Calculator The simplest monomial order is lexicographic. solution is all the values that make true. See more, Polynomial by degree and number of terms calculator, Find the complex zeros of the following polynomial function. Input the roots here, separated by comma. This algebraic expression is called a polynomial function in variable x. What is polynomial equation? $$ \begin{aligned} 2x^2 + 3x &= 0 \\ \color{red}{x} \cdot \left( \color{blue}{2x + 3} \right) &= 0 \\ \color{red}{x = 0} \,\,\, \color{blue}{2x + 3} & \color{blue}{= 0} \\ . Solve Now The coefficients of the resulting polynomial can be calculated in the field of rational or real numbers. This is called the Complex Conjugate Theorem. Webform a polynomial calculator First, we need to notice that the polynomial can be written as the difference of two perfect squares. In a multi-variable polynomial, the degree of a polynomial is the sum of the powers of the polynomial. In this section, we will discuss a variety of tools for writing polynomial functions and solving polynomial equations. Find the zeros of the quadratic function. Substitute \(x=2\) and \(f (-2)=100\) into \(f (x)\). Addition and subtraction of polynomials are two basic operations that we use to increase or decrease the value of polynomials. According to the rule of thumbs: zero refers to a function (such as a polynomial), and the root refers to an equation. Standard form sorts the powers of #x# (or whatever variable you are using) in descending order. 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. Sol. The zero at #x=4# continues through the #x#-axis, as is the case Roots of quadratic polynomial. Further polynomials with the same zeros can be found by multiplying the simplest polynomial with a factor. The zeros of a polynomial calculator can find all zeros or solution of the polynomial equation P (x) = 0 by setting each factor to 0 and solving for x. Polynomial in standard form Polynomials in standard form can also be referred to as the standard form of a polynomial which means writing a polynomial in the descending order of the power of the variable. Polynomial Standard Form Calculator Calculus: Integral with adjustable bounds. Polynomials Calculator This means that if x = c is a zero, then {eq}p(c) = 0 {/eq}. For the polynomial to become zero at let's say x = 1, Notice, at \(x =0.5\), the graph bounces off the x-axis, indicating the even multiplicity (2,4,6) for the zero 0.5. Where. Polynomial in standard form Are zeros and roots the same? A quadratic equation has two solutions if the discriminant b^2 - 4ac is positive. The Factor Theorem is another theorem that helps us analyze polynomial equations. WebThis precalculus video tutorial provides a basic introduction into writing polynomial functions with given zeros. How do you know if a quadratic equation has two solutions? You can change your choice at any time on our, Extended polynomial Greatest Common Divisor in finite field. Write the constant term (a number with no variable) in the end. Form A Polynomial With The Given Zeros Example Problems With Solutions Example 1: Form the quadratic polynomial whose zeros are 4 and 6. The polynomial can be up to fifth degree, so have five zeros at maximum. There are four possibilities, as we can see in Table \(\PageIndex{1}\). Continue to apply the Fundamental Theorem of Algebra until all of the zeros are found. Using factoring we can reduce an original equation to two simple equations. a polynomial function in standard form The solutions are the solutions of the polynomial equation. In the event that you need to form a polynomial calculator Example 3: Write x4y2 + 10 x + 5x3y5 in the standard form. WebFactoring-polynomials.com makes available insightful info on standard form calculator, logarithmic functions and trinomials and other algebra topics. Double-check your equation in the displayed area. Once the polynomial has been completely factored, we can easily determine the zeros of the polynomial. A new bakery offers decorated sheet cakes for childrens birthday parties and other special occasions. The factors of 1 are 1 and the factors of 2 are 1 and 2. Determine all factors of the constant term and all factors of the leading coefficient. function in standard form with zeros calculator Recall that the Division Algorithm. Now we'll check which of them are actual rational zeros of p. Recall that r is a root of p if and only if the remainder from the division of p The calculator writes a step-by-step, easy-to-understand explanation of how the work was done. Then, by the Factor Theorem, \(x(a+bi)\) is a factor of \(f(x)\). A cubic function has a maximum of 3 roots. form The types of polynomial terms are: Constant terms: terms with no variables and a numerical coefficient. Since f(x) = a constant here, it is a constant function. Answer: The zero of the polynomial function f(x) = 4x - 8 is 2. Polynomials Calculator Click Calculate. Polynomials Calculator Polynomial function standard form calculator Thus, all the x-intercepts for the function are shown. We've already determined that its possible rational roots are 1/2, 1, 2, 3, 3/2, 6. 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. Roots calculator that shows steps. At \(x=1\), the graph crosses the x-axis, indicating the odd multiplicity (1,3,5) for the zero \(x=1\). Substitute the given volume into this equation. We solved each of these by first factoring the polynomial and then using the zero factor property on the factored form. Zeros Calculator The first term in the standard form of polynomial is called the leading term and its coefficient is called the leading coefficient. For example: The zeros of a polynomial function f(x) are also known as its roots or x-intercepts. To write polynomials in standard formusing this calculator; 1. Consider this polynomial function f(x) = -7x3 + 6x2 + 11x 19, the highest exponent found is 3 from -7x3. The calculator further presents a multivariate polynomial in the standard form (expands parentheses, exponentiates, and combines similar terms). Find the remaining factors. We can confirm the numbers of positive and negative real roots by examining a graph of the function. The remainder is 25. Standard Form Calculator We can use the Factor Theorem to completely factor a polynomial into the product of \(n\) factors. Look at the graph of the function \(f\) in Figure \(\PageIndex{1}\). If the polynomial is written in descending order, Descartes Rule of Signs tells us of a relationship between the number of sign changes in \(f(x)\) and the number of positive real zeros. We can check our answer by evaluating \(f(2)\). What is the value of x in the equation below? Remember that the domain of any polynomial function is the set of all real numbers. a polynomial function in standard form with Zero To find the other zero, we can set the factor equal to 0. .99 High priority status .90 Full text of sources +15% 1-Page summary .99 Initial draft +20% Premium writer +.91 10289 Customer Reviews User ID: 910808 / Apr 1, 2022 Frequently Asked Questions We find that algebraically by factoring quadratics into the form , and then setting equal to and , because in each of those cases and entire parenthetical term would equal 0, and anything times 0 equals 0. Zeros Calculator There is a similar relationship between the number of sign changes in \(f(x)\) and the number of negative real zeros. But to make it to a much simpler form, we can use some of these special products: Let us find the zeros of the cubic polynomial function f(y) = y3 2y2 y + 2. WebPolynomial Standard Form Calculator - Symbolab New Geometry Polynomial Standard Form Calculator Reorder the polynomial function in standard form step-by-step full pad It is written in the form: ax^2 + bx + c = 0 where x is the variable, and a, b, and c are constants, a 0. For us, the We can use synthetic division to show that \((x+2)\) is a factor of the polynomial. Writing Polynomial Functions With Given Zeros The first one is $ x - 2 = 0 $ with a solution $ x = 2 $, and the second one is $ 2x^2 - 3 = 0 $. WebPolynomial Standard Form Calculator The number 459,608 converted to standard form is 4.59608 x 10 5 Example: Convert 0.000380 to Standard Form Move the decimal 4 places to the right and remove leading zeros to get 3.80 a = Write a polynomial function in standard form with zeros at 0,1, and 2? Write the rest of the terms with lower exponents in descending order. According to the rule of thumbs: zero refers to a function (such as a polynomial), and the root refers to an equation. \(f(x)=\frac{1}{2}x^3+\frac{5}{2}x^22x+10\). Group all the like terms. The calculator converts a multivariate polynomial to the standard form. 2 x 2x 2 x; ( 3) The LibreTexts libraries arePowered by NICE CXone Expertand are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. Sometimes, WebA zero of a quadratic (or polynomial) is an x-coordinate at which the y-coordinate is equal to 0. .99 High priority status .90 Full text of sources +15% 1-Page summary .99 Initial draft +20% Premium writer +.91 10289 Customer Reviews User ID: 910808 / Apr 1, 2022 Frequently Asked Questions Polynomial Roots Calculator Once we have done this, we can use synthetic division repeatedly to determine all of the zeros of a polynomial function. Because \(x =i\) is a zero, by the Complex Conjugate Theorem \(x =i\) is also a zero. By definition, polynomials are algebraic expressions in which variables appear only in non-negative integer powers.In other words, the letters cannot be, e.g., under roots, in the denominator of a rational expression, or inside a function. Here, + =\(\sqrt { 2 }\), = \(\frac { 1 }{ 3 }\) Thus the polynomial formed = x2 (Sum of zeroes) x + Product of zeroes = x2 \(\sqrt { 2 }\)x + \(\frac { 1 }{ 3 }\) Other polynomial are \(\text{k}\left( {{\text{x}}^{\text{2}}}\text{-}\frac{\text{x}}{\text{3}}\text{-1} \right)\) If k = 3, then the polynomial is 3x2 \(3\sqrt { 2 }x\) + 1, Example 5: Find a quadratic polynomial whose sum of zeros and product of zeros are respectively 0,5 Sol. Since 1 is not a solution, we will check \(x=3\). Use synthetic division to evaluate a given possible zero by synthetically dividing the candidate into the polynomial. Sum of the zeros = 4 + 6 = 10 Product of the zeros = 4 6 = 24 Hence the polynomial formed = x 2 (sum of zeros) x + Product of zeros = x 2 10x + 24 This is known as the Remainder Theorem. form Group all the like terms. Solve Now Hence the degree of this particular polynomial is 4. Writing a polynomial in standard form is done depending on the degree as we saw in the previous section. Quadratic Functions are polynomial functions of degree 2. Check. Only positive numbers make sense as dimensions for a cake, so we need not test any negative values. Be sure to include both positive and negative candidates. So we can shorten our list. Use the Rational Zero Theorem to list all possible rational zeros of the function. Precalculus. 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). WebIn math, a quadratic equation is a second-order polynomial equation in a single variable. a rule that determines the maximum possible numbers of positive and negative real zeros based on the number of sign changes of \(f(x)\) and \(f(x)\), \(k\) is a zero of polynomial function \(f(x)\) if and only if \((xk)\) is a factor of \(f(x)\), a polynomial function with degree greater than 0 has at least one complex zero, allowing for multiplicities, a polynomial function will have the same number of factors as its degree, and each factor will be in the form \((xc)\), where \(c\) is a complex number. A polynomial with zeros x=-6,2,5 is x^3-x^2-32x+60=0 in standard form. Standard Form Polynomial 2 (7ab+3a^2b+cd^4) (2ef-4a^2)-7b^2ef Multivariate polynomial Monomial order Variables Calculation precision Exact Result The terms have variables, constants, and exponents. We will use synthetic division to evaluate each possible zero until we find one that gives a remainder of 0. Consider a quadratic function with two zeros, \(x=\frac{2}{5}\) and \(x=\frac{3}{4}\). Function's variable: Examples. Write the rest of the terms with lower exponents in descending order. function in standard form with zeros calculator Let the cubic polynomial be ax3 + bx2 + cx + d x3+ \(\frac { b }{ a }\)x2+ \(\frac { c }{ a }\)x + \(\frac { d }{ a }\)(1) and its zeroes are , and then + + = 0 =\(\frac { -b }{ a }\) + + = 7 = \(\frac { c }{ a }\) = 6 =\(\frac { -d }{ a }\) Putting the values of \(\frac { b }{ a }\), \(\frac { c }{ a }\), and \(\frac { d }{ a }\) in (1), we get x3 (0) x2+ (7)x + (6) x3 7x + 6, Example 8: If and are the zeroes of the polynomials ax2 + bx + c then form the polynomial whose zeroes are \(\frac { 1 }{ \alpha } \quad and\quad \frac { 1 }{ \beta }\) Since and are the zeroes of ax2 + bx + c So + = \(\frac { -b }{ a }\), = \(\frac { c }{ a }\) Sum of the zeroes = \(\frac { 1 }{ \alpha } +\frac { 1 }{ \beta } =\frac { \alpha +\beta }{ \alpha \beta } \) \(=\frac{\frac{-b}{c}}{\frac{c}{a}}=\frac{-b}{c}\) Product of the zeroes \(=\frac{1}{\alpha }.\frac{1}{\beta }=\frac{1}{\frac{c}{a}}=\frac{a}{c}\) But required polynomial is x2 (sum of zeroes) x + Product of zeroes \(\Rightarrow {{\text{x}}^{2}}-\left( \frac{-b}{c} \right)\text{x}+\left( \frac{a}{c} \right)\) \(\Rightarrow {{\text{x}}^{2}}+\frac{b}{c}\text{x}+\frac{a}{c}\) \(\Rightarrow c\left( {{\text{x}}^{2}}+\frac{b}{c}\text{x}+\frac{a}{c} \right)\) cx2 + bx + a, Filed Under: Mathematics Tagged With: Polynomials, Polynomials Examples, ICSE Previous Year Question Papers Class 10, ICSE Specimen Paper 2021-2022 Class 10 Solved, Concise Mathematics Class 10 ICSE Solutions, Concise Chemistry Class 10 ICSE Solutions, Concise Mathematics Class 9 ICSE Solutions, Class 11 Hindi Antra Chapter 9 Summary Bharatvarsh Ki Unnati Kaise Ho Sakti Hai Summary Vyakhya, Class 11 Hindi Antra Chapter 8 Summary Uski Maa Summary Vyakhya, Class 11 Hindi Antra Chapter 6 Summary Khanabadosh Summary Vyakhya, John Locke Essay Competition | Essay Competition Of John Locke For Talented Ones, Sangya in Hindi , , My Dream Essay | Essay on My Dreams for Students and Children, Viram Chinh ( ) in Hindi , , , EnvironmentEssay | Essay on Environmentfor Children and Students in English. Recall that the Division Algorithm. The standard form of a polynomial is given by, f(x) = anxn + an-1xn-1 + an-2xn-2 + + a1x + a0. The solver shows a complete step-by-step explanation. WebPolynomial factoring calculator This calculator is a free online math tool that writes a polynomial in factored form. WebPolynomials Calculator. 95 percent. Explanation: If f (x) has a multiplicity of 2 then for every value in the range for f (x) there should be 2 solutions. Example \(\PageIndex{5}\): Finding the Zeros of a Polynomial Function with Repeated Real Zeros. The coefficients of the resulting polynomial can be calculated in the field of rational or real numbers. Precalculus Polynomial Functions of Higher Degree Zeros 1 Answer George C. Mar 6, 2016 The simplest such (non-zero) polynomial is: f (x) = x3 7x2 +7x + 15 Explanation: As a product of linear factors, we can define: f (x) = (x +1)(x 3)(x 5) = (x +1)(x2 8x + 15) = x3 7x2 +7x + 15 Polynomial Function Real numbers are also complex numbers. a) f(x) = x1/2 - 4x + 7 b) g(x) = x2 - 4x + 7/x c) f(x) = x2 - 4x + 7 d) x2 - 4x + 7. In this article, we will learn how to write the standard form of a polynomial with steps and various forms of polynomials. 6x - 1 + 3x2 3. x2 + 3x - 4 4. Legal. In this example, the last number is -6 so our guesses are. Although I can only afford the free version, I still find it worth to use. This theorem forms the foundation for solving polynomial equations. Note that this would be true for f (x) = x2 since if a is a value in the range for f (x) then there are 2 solutions for x, namely x = a and x = + a. Zeros of a Polynomial Function Example \(\PageIndex{7}\): Using the Linear Factorization Theorem to Find a Polynomial with Given Zeros. Example 3: Find a quadratic polynomial whose sum of zeros and product of zeros are respectively\(\frac { 1 }{ 2 }\), 1 Sol. The zeros are \(4\), \(\frac{1}{2}\), and \(1\). "Poly" means many, and "nomial" means the term, and hence when they are combined, we can say that polynomials are "algebraic expressions with many terms". A complex number is not necessarily imaginary. This means that, since there is a \(3^{rd}\) degree polynomial, we are looking at the maximum number of turning points. However, when dealing with the addition and subtraction of polynomials, one needs to pair up like terms and then add them up. Precalculus. Zeros Calculator Calculator shows detailed step-by-step explanation on how to solve the problem. Here, a n, a n-1, a 0 are real number constants. \[f(\dfrac{1}{2})=2{(\dfrac{1}{2})}^3+{(\dfrac{1}{2})}^24(\dfrac{1}{2})+1=3\]. This algebraic expression is called a polynomial function in variable x. Example 3: Find the degree of the polynomial function f(y) = 16y5 + 5y4 2y7 + y2. Example \(\PageIndex{6}\): Finding the Zeros of a Polynomial Function with Complex Zeros. To find its zeros, set the equation to 0. We can conclude if \(k\) is a zero of \(f(x)\), then \(xk\) is a factor of \(f(x)\). Polynomial function in standard form calculator Polynomial in standard form Roots of quadratic polynomial. Check. WebZero: A zero of a polynomial is an x-value for which the polynomial equals zero. Before we give some examples of writing numbers in standard form in physics or chemistry, let's recall from the above section the standard form math formula:. WebZero: A zero of a polynomial is an x-value for which the polynomial equals zero. Standard Form Calculator Example 04: Solve the equation $ 2x^3 - 4x^2 - 3x + 6 = 0 $. If the remainder is 0, the candidate is a zero. How do you know if a quadratic equation has two solutions?

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