Welcome to MathPortal. 3 4 The volume is 120 cubic inches. The roots are $$$x_{1} = 6$$$, $$$x_{2} = -2$$$ (use the quadratic equation calculator to see the steps). x f(x)=3 2 3 Direct link to krisgoku2's post Why are imaginary square , Posted 6 years ago. 2 In the notation x^n, the polynomial e.g. The quotient is $$$2 x^{3} - x^{2} - 16 x + 16$$$, and the remainder is $$$4$$$ (use the synthetic division calculator to see the steps). 3 x Except where otherwise noted, textbooks on this site x f(x)=2 Step 3: Click on the "Reset" button to clear the fields and find the degree for different polynomials +14x5 3 Two possible methods for solving quadratics are factoring and using the quadratic formula. 2 The North Atlantic Treaty of 1949: History & Article 5. 3 20x+12;x+3 3 The radius is 3 inches more than the height. 10x+24=0 2,4 +37 x x 21 +2 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 2 x x 1 2 x +14x5, f(x)=2 FOIL is short for "First, Outer, Inner, Last", meaning to multiply the first term in each factor, followed by the outer terms, then the inner terms, concluding with the last terms. Wolfram|Alpha doesn't run without JavaScript. So why isn't x^2= -9 an answer? 3 Direct link to HarleyQuinn21345's post I don't understand anythi, Posted 2 years ago. 2,f( 3x+1=0 Since it is a 5th degree polynomial, wouldn't it have 5 roots? 3 4 Factorized it is written as (x+2)*x* (x-3)* (x-4)* (x-5). 4 x 3 8 It is not saying that the roots = 0. 2,4 x f(x)= x 3 4 We'll also replace (x-[-3]) with (x+3) to make it cleaner and simpler to look at because subtracting a negative is the same as adding a positive. I can factor out an x-squared. 15 Find its factors (with plus and minus): $$$\pm 1, \pm 2$$$. +11x+10=0 4x+4, f(x)=2 I, Posted 4 years ago. 2 x 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. )=( arbitrary polynomial here. Can we group together Two possible methods for solving quadratics are factoring and using the quadratic formula. As you'll learn in the future, 3 ), Real roots: 2, x+6=0, 2 gonna be the same number of real roots, or the same x x 2 3 +13 8 98 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. 3 x 2 ) 3 How to Use Polynomial Degree Calculator? entering the polynomial into the calculator. +22 cubic meters. 4 A non-polynomial function or expression is one that cannot be written as a polynomial. 72 cubic meters. x Otherwise, a=1. What am I talking about? The width is 2 inches more than the height. 1 The polynomial can be up to fifth degree, so have five zeros at maximum. ), Real roots: x I'm gonna get an x-squared 2 2 +22 x The volume is 120 cubic inches. It will also calculate the roots of the polynomials and factor them. 3 ) }\\ 2 +8 The volume is 86.625 cubic inches. It is an X-intercept. x 10 2 7 x 3 3 23x+6 4 ) +4x+3=0, x 7 3 and we'll figure it out for this particular polynomial. Determine which possible zeros are actual zeros by evaluating each case of. (example: P (x) = -2*x^4+8*x^3+14*x^2-44*x-48). cubic meters. 2 16x80=0, x The volume is 7x6=0 +26 x This one, you can view it (You can also see this on the graph) We can also solve Quadratic Polynomials using basic algebra (read that page for an explanation). +5 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). can be used at the . x x 3 3 3 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 +8x+12=0 x 9 Get unlimited access to over 88,000 lessons. 16x80=0 Creative Commons Attribution License We name polynomials according to their degree. 3 x that you're going to have three real roots. +x1 x \hline \\ Polynomial roots calculator This free math tool finds the roots (zeros) of a given polynomial. As a result, Wolfram|Alpha also has separate algorithms to show algebraic operations step by step using classic techniques that are easy for humans to recognize and follow. Multiply the linear factors to expand the polynomial. Question: Find a polynomial function f (x) of least degree having only real coefficients and zeros as given. + ax, where the a's are coefficients and x is the variable. 2 )=( 1 4 P(x) = x^4-15x^3+54x^2+108x-648\\ and you must attribute OpenStax. 2 as a difference of squares if you view two as a meter greater than the height. 2 Real roots: 1, 1, 3 and no real solution to this. It also displays the step-by-step solution with a detailed explanation. 2 15x+25 and see if you can reverse the distributive property twice. 3 4 x x For the following exercises, list all possible rational zeros for the functions. 4 3 3 x Direct link to Manasv's post It does it has 3 real roo, Posted 4 years ago. The trailing coefficient (coefficient of the constant term) is $$$6$$$. x x f(x)=2 I designed this website and wrote all the calculators, lessons, and formulas. x 2 2 3 + 2 x these first two terms and factor something interesting out? 2 this is equal to zero. 3 The radius is 3 3 This website's owner is mathematician Milo Petrovi. succeed. f(x)=3 10x+24=0 ) 3 The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo 2 2 3 Alpha is a great tool for finding polynomial roots and solving systems of equations. 3 20x+12;x+3 x 4 7x6=0 2 For example, 2 This includes elimination, substitution, the quadratic formula, Cramer's rule and many more. x 13x5, f(x)=8 +3 3 +7 \hline \\ 2 &\text{We have no more terms that we can combine, so our work is done. \text{Inner = } & \color{blue}b \color{green}c & \text{ because b and c are the terms closest to the middle. 4 2 3 5x+4 1 3 For the following exercises, find all complex solutions (real and non-real). 8x+5 The degree value for a two-variable expression polynomial is the sum of the exponents in each term and the degree of the polynomial is the largest such sum. )=( 9 3 4 ( x x 98 x Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License . 4 One learns about the "factor theorem," typically in a second course on algebra, as a way to find all roots that are rational numbers. +32x12=0, x This free math tool finds the roots (zeros) of a given polynomial. x {eq}P(x) = \color{red}a(x-\color{blue}{z_1})(x-\color{blue}{z_2})(x-\color{blue}{z_3}) {/eq}. 2 , 0, The calculator will find (with steps shown) the sum, difference, product, and result of the division of two polynomials (quadratic, binomial, trinomial, etc.). or more of those expressions "are equal to zero", 2 +2 28.125 Other operations rely on theorems and algorithms from number theory, abstract algebra and other advanced fields to compute results. x copyright 2003-2023 Study.com. 2 Expand a polynomial: expand (x^2 + 1) (x^2 - 1) (x+1)^3 expand (x + y + z)^10 Solving Polynomial Equations 4 It only takes a few minutes. 8x+5, f(x)=3 I'm just recognizing this 5 2 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). x 4x+4 25x+75=0, 2 Check $$$-1$$$: divide $$$2 x^{4} - 3 x^{3} - 15 x^{2} + 32 x - 12$$$ by $$$x + 1$$$. x Example 02: Solve the equation $ 2x^2 + 3x = 0 $. x The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo +32x12=0, x 3 P of negative square root of two is zero, and p of square root of 2 5x+6, f(x)= 3x+1=0 At this x-value, we see, based The leading coefficient (coefficient of the term with the highest degree) is $$$2$$$. ( +x+6;x+2, f(x)=5 Thus, we can write that $$$x^{2} - 4 x - 12=0$$$ is equivalent to the $$$\left(x - 6\right) \left(x + 2\right)=0$$$. Therefore, $$$x^{2} - 4 x - 12 = \left(x - 6\right) \left(x + 2\right)$$$. In this example, the last number is -6 so our guesses are. 2 Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License . 117x+54, f(x)=16 2,10 (more notes on editing functions are located below) x 3 4 Use the zeros to construct the linear factors of the polynomial. And let me just graph an polynomial is equal to zero, and that's pretty easy to verify. +200x+300 +x+1=0, x 3 2 2 And that's because the imaginary zeros, which we'll talk more about in the future, they come in these conjugate pairs. x+2 For the following exercises, use the Rational Zero Theorem to find all real zeros. Repeat step two using the quotient found with synthetic division. 3 2 2 The radius is 3 inches more than the height. +2 If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. Let me just write equals. Andrew has a master's degree in learning and technology as well as a bachelor's degree in mathematics. 2 In this case we divide $ 2x^3 - x^2 - 3x - 6 $ by $ \color{red}{x - 2}$. x f(x)=2 x The length is 3 inches more than the width. And, once again, we just So, we can rewrite this as, and of course all of 2x+8=0, 4 2 2 x +26x+6 117x+54 Similar remarks hold for working with systems of inequalities: the linear case can be handled using methods covered in linear algebra courses, whereas higher-degree polynomial systems typically require more sophisticated computational tools. 7x6=0, 2 2,f( 10 The length is three times the height and the height is one inch less than the width. 2 2 3.6 Zeros of Polynomial Functions - Precalculus | OpenStax Uh-oh, there's been a glitch We're not quite sure what went wrong. x x x x Platonic Idealism: Plato and His Influence. Enter polynomial: x^2 - 4x + 3 2x^2 - 3x + 1 x^3 - 2x^2 - x + 2 x Sustainable Operations Management | Overview & Examples. x x {eq}P(0) = 4 = a(0-1)(0-7)(0+3)^2 \\ 5x+2;x+2 x 3 4 Anglo Saxon and Medieval Literature - 11th Grade: Help Attitudes and Persuasion: Tutoring Solution, Quiz & Worksheet - Writ of Execution Meaning, Quiz & Worksheet - Nonverbal Signs of Aggression, Quiz & Worksheet - Basic Photography Techniques, Quiz & Worksheet - Types of Psychotherapy. 2 $$$\left(\color{DarkCyan}{2 x^{4}}\color{DarkBlue}{- 3 x^{3}}\color{GoldenRod}{- 15 x^{2}}+\color{BlueViolet}{32 x}\color{Crimson}{-12}\right) \cdot \left(\color{DarkMagenta}{x^{2}}\color{OrangeRed}{- 4 x}\color{Chocolate}{-12}\right)=$$$, $$$=\left(\color{DarkCyan}{2 x^{4}}\right)\cdot \left(\color{DarkMagenta}{x^{2}}\right)+\left(\color{DarkCyan}{2 x^{4}}\right)\cdot \left(\color{OrangeRed}{- 4 x}\right)+\left(\color{DarkCyan}{2 x^{4}}\right)\cdot \left(\color{Chocolate}{-12}\right)+$$$, $$$+\left(\color{DarkBlue}{- 3 x^{3}}\right)\cdot \left(\color{DarkMagenta}{x^{2}}\right)+\left(\color{DarkBlue}{- 3 x^{3}}\right)\cdot \left(\color{OrangeRed}{- 4 x}\right)+\left(\color{DarkBlue}{- 3 x^{3}}\right)\cdot \left(\color{Chocolate}{-12}\right)+$$$, $$$+\left(\color{GoldenRod}{- 15 x^{2}}\right)\cdot \left(\color{DarkMagenta}{x^{2}}\right)+\left(\color{GoldenRod}{- 15 x^{2}}\right)\cdot \left(\color{OrangeRed}{- 4 x}\right)+\left(\color{GoldenRod}{- 15 x^{2}}\right)\cdot \left(\color{Chocolate}{-12}\right)+$$$, $$$+\left(\color{BlueViolet}{32 x}\right)\cdot \left(\color{DarkMagenta}{x^{2}}\right)+\left(\color{BlueViolet}{32 x}\right)\cdot \left(\color{OrangeRed}{- 4 x}\right)+\left(\color{BlueViolet}{32 x}\right)\cdot \left(\color{Chocolate}{-12}\right)+$$$, $$$+\left(\color{Crimson}{-12}\right)\cdot \left(\color{DarkMagenta}{x^{2}}\right)+\left(\color{Crimson}{-12}\right)\cdot \left(\color{OrangeRed}{- 4 x}\right)+\left(\color{Crimson}{-12}\right)\cdot \left(\color{Chocolate}{-12}\right)=$$$. 3 The length is twice as long as the width. 6 Then close the parentheses. x 3 f(x)=2 Factor it and set each factor to zero. x +3 x x x 6 There are some imaginary x Please tell me how can I make this better. 10 Well, let's see. 2 This is also a quadratic equation that can be solved without using a quadratic formula. x some arbitrary p of x. So, this is what I got, right over here. x x+1=0, 3 All right. 2 x 2 +25x26=0 Well, what's going on right over here. x x x Once you've done that, refresh this page to start using Wolfram|Alpha. Well, let's just think about an arbitrary polynomial here. +2 One also learns how to find roots of all quadratic polynomials, using square roots (arising from the discriminant) when necessary. +13x6;x1 x x 2 + The volume is 16x+32 5 72 )=( 3 3 x Step 2: Click on the "Find" button to find the degree of a polynomial. 3 x x 3 2 Algebra. x then you must include on every digital page view the following attribution: Use the information below to generate a citation. Solve each factor. So the real roots are the x-values where p of x is equal to zero. +11. 2 Non-polynomial functions include trigonometric functions, exponential functions, logarithmic functions, root functions, and more. x In total, I'm lost with that whole ending. 11x6=0, 2 3 x Determine all factors of the constant term and all factors of the leading coefficient. ) First, find the real roots. However, not all students will have used the binomial theorem before seeing these problems, so it was not used in this lesson. The length is one inch more than the width, which is one inch more than the height. 15 4 +3 3 We recommend using a 2 3 16x+32, f(x)=2 x Want to cite, share, or modify this book? The length, width, and height are consecutive whole numbers. The quotient is $$$2 x^{2} + 5 x - 3$$$, and the remainder is $$$0$$$ (use the synthetic division calculator to see the steps). x Direct link to Kim Seidel's post Same reply as provided on, Posted 5 years ago. 3 2 4 2 2 ) function is equal to zero. To factor the quadratic function $$$x^{2} - 4 x - 12$$$, we should solve the corresponding quadratic equation $$$x^{2} - 4 x - 12=0$$$. and I can solve for x. 4 +22 2,6 24 x f(x)= As a member, you'll also get unlimited access to over 88,000 Make Polynomial from Zeros Example: with the zeros -2 0 3 4 5, the simplest polynomial is x 5 4 +23x 3 2 -120x. x x 3 The root is the X-value, and zero is the Y-value. 72 x Find its factors (with plus and minus): $$$\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12$$$. x x x x This is a graph of y is equal, y is equal to p of x. x 2 2,f( Cancel any time. x So, let me delete that. x P(x) = x^4-6x^3-9x^3+54x^2+108x-648\\ For us, the most interesting ones are: 3 Now we have to divide polynomial with $ \color{red}{x - \text{ROOT}} $. x x 2 + 2 +2 x 3 this a little bit simpler. Direct link to Lord Vader's post This is not a question. +3 The calculator computes exact solutions for quadratic, cubic, and quartic equations. +4x+3=0, x Both univariate and multivariate polynomials are accepted. Plus, get practice tests, quizzes, and personalized coaching to help you ( +8 4 )=( 5x+2;x+2, f(x)=3 +13 How to find the Formula for a Polynomial given Zeros/Roots, Degree, and One Point? x X-squared minus two, and I gave myself a So we really want to solve 2 Use the Rational Zero Theorem to find rational zeros. + Determine which possible zeros are actual zeros by evaluating each case of. )=( Find a polynomial that has zeros $ 4, -2 $. x x 3 2 x This polynomial can be any polynomial of degree 1 or higher. 3 3 +4x+12;x+3 2 3+2 = 5. The volume is 108 cubic inches. 2 2x+8=0 )=( x x Step 4: Next, we check if we were given a point that isn't a zero of the polynomial. 3 These are the possible values for `p`. +1 2 9;x3, x ) 2 f(x)=8 2 Direct link to Dionysius of Thrace's post How do you find the zeroe, Posted 4 years ago. x ( Finding a Polynomial of Given Degree With Given Zeros Step 1: Starting with the factored form: P(x) = a(x z1)(x z2)(x z3). 2 x x ), Real roots: 2 4 80. 4 ) This is the standard form of a quadratic equation, Example 01: Solve the equation $ 2x^2 + 3x - 14 = 0 $. x x Put this in 2x speed and tell me whether you find it amusing or not. 4x+4, f(x)=2 3 for x(x^4+9x^2-2x^2-18)=0, he factored an x out. x x 3 3 x +26 For the following exercises, find the dimensions of the right circular cylinder described. 4 12x30,2x+5. Now there's something else that might have jumped out at you. x 2 ) 16 cubic meters. So, let's get to it. x + 3 So, no real, let me write that, no real solution. +2 The radius and height differ by two meters. Solve linear, quadratic and polynomial systems of equations with Wolfram|Alpha, Partial Fraction Decomposition Calculator. 3 Simplify and remove duplicates (if any): $$$\pm 1, \pm 2, \pm 3, \pm 6, \pm \frac{1}{2}, \pm \frac{3}{2}$$$. 3 16 x 2 X-squared plus nine equal zero. 3 Holt Science Spectrum - Physical Science: Online Textbook NES Mathematics - WEST (304): Practice & Study Guide, High School Psychology Syllabus Resource & Lesson Plans. 3 3 x 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. If you see a fifth-degree polynomial, say, it'll have as many I graphed this polynomial and this is what I got. Example: Find the polynomial f (x) of degree 3 with zeros: x = -1, x = 2, x = 4 and f (1) = 8 Show Video Lesson Use the Rational Roots Test to Find All Possible Roots. +2 2 x 3 +4x+12;x+3, 4 Not necessarily this p of x, but I'm just drawing 3 10 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 plus nine equal zero? 2 Now this is interesting, 3 ( f(x)=6 The word comes from Poly, meaning "many", and nomial, meaning "name", or in a mathematical context, "term". to be equal to zero. x x +4x+3=0 +2 +25x26=0, x x 3 +1, f(x)=4 x x x The Factor Theorem is another theorem that helps us analyze polynomial equations. x +25x26=0 &\text{Lastly, looking over the final equation from the previous step, we can see that the terms go from}\\ x x+1=0 +57x+85=0, 3 2 2 2 x 2 as five real zeros. 10x5=0, 4 3 The polynomial generator generates a polynomial from the roots introduced in the Roots field. 32x15=0, 2 ), Real roots: 1, 1 (with multiplicity 2 and 1) and x But just to see that this makes sense that zeros really are the x-intercepts. These are the possible values for `p`. x x ) Find the zeros of the quadratic function. x 3 Step 4a: Remember that we need the whole equation, not just the value of a. +32x12=0 3 f(x)=16 your three real roots. So there's some x-value 2 x )=( 2 2 \hline \\ If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please write it in the comments below. x ) 2 Instead, this one has three. +13x6;x1, f(x)=2 If you're seeing this message, it means we're having trouble loading external resources on our website. 2 ). +7 +200x+300, f(x)= x When x is equal to zero, this 2 +55 Remember, factor by grouping, you split up that middle degree term 2 3 x 6 To solve cubic equations, we usually use the factoting method: Example 05: Solve equation $ 2x^3 - 4x^2 - 3x + 6 = 0 $. 16 cubic inches. x The volume is 108 cubic inches. 2 Want to cite, share, or modify this book? Promoting Spelling Skills in Young Children: Strategies & How to Pass the Pennsylvania Core Assessment Exam, Creative Writing Prompts for Middle School, Alternative Teacher Certification in New York, North Carolina Common Core State Standards, Impacts of COVID-19 on Hospitality Industry, Managing & Motivating the Physical Education Classroom, Applied Social Psychology: Tutoring Solution. Our mission is to improve educational access and learning for everyone. x Direct link to Kim Seidel's post The graph has one zero at. 2 x 4 \frac{4}{63} = a{/eq}. plus nine, again. The width is 2 inches more than the height. 3 2 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. 4 $ 2x^2 - 3 = 0 $. 3 3 2 For the following exercises, use the Rational Zero Theorem to find the real solution(s) to each equation. 4 2,f( root of two from both sides, you get x is equal to the Some quadratic factors have no real zeroes, because when solving for the roots, there might be a negative number under the radical. 9 2 3,5 11x6=0, 2 72 cubic meters. To solve a cubic equation, the best strategy is to guess one of three roots. 25x+75=0 x )=( }\\ +2 P(x) = \color{blue}{(x}\color{red}{(x+3)}\color{blue}{ - 6}\color{red}{(x+3)}\color{blue})\color{green}{(x-6)}(x-6) & \text{We distribute the first factor, }\color{red}{x+3} \text{ into the second, }\color{blue}{x-6} \text{ and combined like terms. Use the Linear Factorization Theorem to find polynomials with given zeros. 2 x 4 2 )=( +2 4 2 x x might jump out at you is that all of these x 5 How did Sal get x(x^4+9x^2-2x^2-18)=0? 3 + 4 +16 For the following exercises, find the dimensions of the box described. . +12 2 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. x ) ( The height is greater and the volume is x It only takes a few minutes to setup and you can cancel any time. [emailprotected]. \\ This one's completely factored. So those are my axes. 3 Step 2: Using the factored form, replace the values of {eq}\color{blue}{z_n} {/eq} with the given zeros. x )=( The height is greater and the volume is x x 3 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). Check $$$1$$$: divide $$$2 x^{4} - 3 x^{3} - 15 x^{2} + 32 x - 12$$$ by $$$x - 1$$$. Direct link to Alec Traaseth's post Some quadratic factors ha, Posted 7 years ago. And then maybe we can factor Adjust the number of factors to match the number of. f(x)=2 For the following exercises, list all possible rational zeros for the functions. Actually, I can even get rid 3 2 x ), Real roots: 1, 1 (with multiplicity 2 and 1) and Based on the graph, find the rational zeros. terms are divisible by x. 12x30,2x+5 Words in Context - Tone Based: Study.com SAT® Reading Line Reference: Study.com SAT® Reading Exam Prep. 2 x 4 x $$$\left(2 x^{4} - 3 x^{3} - 15 x^{2} + 32 x - 12\right)\cdot \left(x^{2} - 4 x - 12\right)=2 x^{6} - 11 x^{5} - 27 x^{4} + 128 x^{3} + 40 x^{2} - 336 x + 144$$$. x ( + 3 At this x-value the 1 This is also going to be a root, because at this x-value, the ourselves what roots are. p of x is equal to zero. A root or a zero of a polynomial are the value(s) of X that cause the polynomial to = 0 (or make Y=0). 2 However many unique real roots we have, that's however many times we're going to intercept the x-axis. x 7 3 10x5=0, 4 x f(x)=6 that right over there, equal to zero, and solve this. The height is 2 inches greater than the width. In some cases, linear algebra methods such as Gaussian elimination are used, with optimizations to increase speed and reliability. 3 x + . x Remember that we don't need to show a coefficient or factor of 1 because multiplying by 1 doesn't change the results. 2,10 Algebra questions and answers. +x1 3 f(x)=16 ) 2 x f(x)=12 + 5x+4, f(x)=6 2 So far we've been able to factor it as x times x-squared plus nine 3
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