rotate. A parabola is graphed on an x y coordinate plane. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. Let's understand this with the polynomial, When a linear factor occurs multiple times in the factorization of a polynomial, that gives the related zero. Odd Negative Graph goes zero when x is equal to 3/2. What if you have a funtion like f(x)=-3^x? What about functions like, In general, the end behavior of a polynomial function is the same as the end behavior of its, This is because the leading term has the greatest effect on function values for large values of, Let's explore this further by analyzing the function, But what is the end behavior of their sum? Off topic but if I ask a question will someone answer soon or will it take a few days? You don't have to know this to solve the problem. I don't see an x minus 3/2 here, but as we've mentioned in other videos you can also multiply sinusoidal functions will repeat till infinity unless you restrict them to a domain. http://cnx.org/contents/9b08c294-057f-4201-9f48-5d6ad992740d@5.2. WebWrite an equation for the polynomial graphed below. The graph curves up from left to right touching (one, zero) before curving down. Excellent App, the application itself is great for a wide range of math levels, i don't have to wait for memo to check my answers if they are correct and it is very helpful as it explains ever steps that lead to solution. (Say, "as x x approaches positive infinity, f (x) f (x) approaches positive infinity.") Let's algebraically examine the end behavior of several monomials and see if we can draw some conclusions. Direct link to Danish Anwar's post how did u get 3/2, Posted 6 months ago. A cubic function is graphed on an x y coordinate plane. Write an equation for the polynomial graphed below. minus 3/2 in our product. equal to negative four, we have a zero because our Sometimes, a turning point is the highest or lowest point on the entire graph. a) What percentage of years will have an annual rainfall of less than 44 inches? 2. polynomial p right over here, you could view this as the graph of y is equal to p of x. Solution for Write an equation for the polynomial graphed below with degree 4. graph is attached as jpg. Transcribed Image Text:Write an equation for the polynomial graphed below 5+ 4- 2. We can also determine the end behavior of a polynomial function from its equation. Direct link to 335697's post Off topic but if I ask a , Posted a year ago. Mathematics is the study of numbers, shapes and patterns. to see the solution. WebWrite the equation of a polynomial function given its graph. in total there are 3 roots as we see in the equation . Polynomial From Roots Generator input roots 1/2,4 and calculator will generate a polynomial show help examples Enter roots: display polynomial graph Generate Polynomial examples example 1: It depends on the job that you want to have when you are older. Quality is important in all aspects of life. Posted 7 years ago. Find the polynomial of least degree containing all of the factors found in the previous step. when x is equal to three, and we indeed have that right over there. So I'm liking choices B and D so far. Using technology to sketch the graph of [latex]V\left(w\right)[/latex] on this reasonable domain, we get a graph like the one above. Direct link to ReignDog2's post I was wondering how this , Posted 2 years ago. A global maximum or global minimum is the output at the highest or lowest point of the function. Each turning point represents a local minimum or maximum. WebFinding the x -Intercepts of a Polynomial Function Using a Graph Find the x -intercepts of h(x) = x3 + 4x2 + x 6. The x-axis scales by one. Review How to Find the Equations of a Polynomial Function from its Graph in this Precalculus tutorial. Direct link to devarakonda balraj's post how to find weather the g, Posted 6 years ago. I need so much help with this. is equal to negative four, we probably want to have a term that has an x plus four in it. If a function has a global maximum at a, then [latex]f\left(a\right)\ge f\left(x\right)[/latex] for all x. and standard deviation 5.3 inches. Reliable Support is a company that provides quality customer service. Direct link to Kim Seidel's post Questions are answered by, Posted 2 years ago. At x= 3 and x= 5,the graph passes through the axis linearly, suggesting the corresponding factors of the polynomial will be linear. - [Instructor] We are asked, what could be the equation of p? what is the polynomial remainder theorem? So if I were to multiply, let's see to get rid The graph curves down from left to right touching (negative four, zero) before curving up. Direct link to THALIA GRACE's post how does the point: 1.5 m, Posted 2 years ago. For now, we will estimate the locations of turning points using technology to generate a graph. A "passing grade" is a grade that is good enough to get a student through a class or semester. What are the end behaviors of sine/cosine functions? To determine the stretch factor, we utilize another point on the graph. Write an equation for the 4th degree polynomial graphed below. why the power of a polynomial can not be negative or in fraction? WebBelow are graphs, grouped according to degree, showing the different sorts of "bump" collection each degree value, from two to six, can have. Polynomial Function Graph. To graph a simple polynomial function, we usually make a table of values with some random values of x and the corresponding values of f(x). Then we plot the points from the table and join them by a curve. Let us draw the graph for the quadratic polynomial function f(x) = x 2. If you're looking for help with your studies, our expert tutors can give you the guidance you need to succeed. How to find 4th degree polynomial equation from given points? On the graph of a function, the roots are the values of x for which it crosses the x-axis, hence they are given as follows: When x = 0, y = -3, hence the leading coefficient a is found as follows: More can be learned about the Factor Theorem at brainly.com/question/24380382, This site is using cookies under cookie policy . Obviously, once you get to math at this stage, only a few jobs use them. Given the graph below, write a formula for the function shown. The expression for the polynomial graphed will be y(x) = (x + 3)(x - 1 )(x - 4 ). When x is equal to negative four, this part of our product is equal to zero which makes the Do all polynomial functions have a global minimum or maximum? The middle of the parabola is dashed. Can someone please explain what exactly the remainder theorem is? If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. To log in and use all the features of Khan Academy, please enable JavaScript in your browser. Use k if your leading coefficient is positive and -k if your leading coefficient is negative. WebWrite an equation for the polynomial graphed below. You'll get a, VIDEO ANSWER: So in this problem, what they want us to do is to write an equation for the polynomial graph below. Make sure to observe both positive and negative [latex]a[/latex]-values, and large and small [latex]a[/latex]-values. The solutions to the linear equations are the zeros of the polynomial function. The shortest side is 14 and we are cutting off two squares, so values wmay take on are greater than zero or less than 7. [latex]\begin{array}{l}f\left(0\right)=a\left(0+3\right){\left(0 - 2\right)}^{2}\left(0 - 5\right)\hfill \\ \text{ }-2=a\left(0+3\right){\left(0 - 2\right)}^{2}\left(0 - 5\right)\hfill \\ \text{ }-2=-60a\hfill \\ \text{ }a=\frac{1}{30}\hfill \end{array}[/latex]. d2y. dt2. + n2y = 0. whose general solution is. y = A cos nt + B sin nt. or as. |x| < 1. or equivalently. y = ATn (x) + BUn (x) |x| < 1. where Tn (x) and Un (x) are defined as Chebyshev polynomials of the first and second kind. of degree n, respectively. For example: f(x)=(x+3)^2+(x-5)(x-3)^-1, how to find weather the graph is (odd or even). Given: The graph of the polynomial is shown below: From the above graph, it can be observed that there are four x x intercepts at x=-3,x=-2,x=1andx=3 x. Compare the numbers of bumps in the graphs below to the degrees of their to make some intelligent guesses about polynomials from their graphs, and about Deal with mathematic problems. please help me . With a constant term, things become a little more interesting, because the new function actually isn't a polynomial anymore. With quadratics, we were able to algebraically find the maximum or minimum value of the function by finding the vertex. And we have graph of our From this zoomed-in view, we can refine our estimate for the maximum volume to about 339 cubic cm which occurs when the squares measure approximately 2.7 cm on each side. I'm grateful enough that I even have the opportunity to have such a nice education compared to developing countries where most citizens never make it to college. Algebra questions and answers. Well we have an x plus four there, and we have an x plus four there. I think it's a very needed feature, a great calculator helps with all math and geometry problems and if you can't type it you can take a picture of it, super easy to use and great quality. WebQuestion: Write an equation for the polynomial graphed below Show transcribed image text Expert Answer Transcribed image text: Write an equation for the polynomial graphed Direct link to aasthanhg2e's post what is the polynomial re, Posted a year ago. In challenge problem 8, I don't know understand how we get the general shape of the graph, as in how do we know when it continues in the positive or negative direction. Webwrite an equation for the polynomial graphed below Given: The graph of the polynomial is shown below: From the above graph, it can be observed that there are four x x intercepts at x=-3,x=-2,x=1andx=3 x If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. entire product equal to zero. 4x + 5x - 12 What is the Factor Theorem? Direct link to Raquel Ortiz's post Is the concept of zeros o, Posted 2 years ago. % Identifying Zeros and Their Multiplicities Graphs behave differently at various x Quite simple acutally. Direct link to Laila B. f_f(x)=4x^5-5x^3 , but also f_f(x)=3 Solve Now The graph curves down from left to right passing through the origin before curving down again. We also know that p of, looks like 1 1/2, or I could say 3/2. And let's see, we have a two x It gives vivid method and understanding to basic math concept and questions. In this article, we will explore these characteristics of polynomials and the special relationship that they have with each other. Web47.1. WebGiven: The graph of the polynomial is shown below: From the above graph, it can be observed that there are four x x intercepts at x=-3,x=-2,x=1andx=3 x Convert standard form to slope intercept form, How are radical expressions & rational exponents used in real life, How to find domain and range of a relation on a graph, Jobs you can get with applied mathematics. The annual rainfall in a certain region is approximately normally distributed with mean 40.9 inches it with this last one. Using multiplity how can you find number of real zeros on a graph. Direct link to shub112's post Using multiplity how can , Posted 3 years ago. Math is a way of solving problems by using numbers and equations. Use k if your leading coefficient is positive and-k if your leading coefficlent Fourth Degree Polynomials. Find a Polynomial Function From a Graph w/ Least Possible Degree | Linear Factors, Adding and subtracting fractions review worksheet, Factor quadratic equations into two binomials, Factorization of algebraic expressions questions, Find the degree of each monomial calculator, Find three consecutive integers that have a sum of 96, How to find the difference of two squares, How to subtract exponents with different exponents, Solving linear diophantine equations two variables, Transforming linear functions worksheet answers algebra 2. No matter what else is going on in your life, always remember to stay focused on your job. Thank you for trying to help me understand. At x= 2, the graph bounces off the x-axis at the intercept suggesting the corresponding factor of the polynomial will be second degree (quadratic). To log in and use all the features of Khan Academy, please enable JavaScript in your browser. We will start this problem by drawing a picture like the one below, labeling the width of the cut-out squares with a variable, w. Notice that after a square is cut out from each end, it leaves a [latex]\left(14 - 2w\right)[/latex] cm by [latex]\left(20 - 2w\right)[/latex] cm rectangle for the base of the box, and the box will be wcm tall. The question asks about the multiplicity of the root, not whether the root itself is odd or even. Direct link to rylin0403's post Quite simple acutally. Direct link to kyle.davenport's post What determines the rise , Posted 5 years ago. Well, let's start with a positive leading coefficient and an even degree. It curves back down and passes through (six, zero). two x minus three is equal to zero which makes the When x is equal to negative four, this part of our product is equal to zero which makes the whole thing equal to zero. Think about the function's graph. Direct link to 999988024's post Hi, How do I describe an , Posted 3 years ago. Solve the equations from Step 1. I have been using it for years and it helped me everytime, whether it was for an exam or just plain entertainment, this app is honesty really great and easy to use i would definitely recommend it. If the value of the coefficient of the term with the greatest degree is positive then that means that the end behavior to on both sides. Precalculus Help Polynomial Functions Graphs of Polynomial Functions Write the Equation of a Polynomial Function Based on Its Graph. Sometimes, roots turn out to be the same (see discussion above on "Zeroes & Multiplicity"). Direct link to A/V's post Typically when given only, Posted 2 years ago. at the "ends. in the answer of the challenge question 8 how can there be 2 real roots . For any polynomial graph, the number of distinct. That refers to the output of functions p, just like f(x) is the output of function f. Function p takes in an input of x, and then does something to it to create p(x). Get math help online by speaking to a tutor in a live chat. What if there is a problem like (x-1)^3 (x+2)^2 will the multiplicity be the addition of 3 and 2 or the highest exponent will be the multiplicity? WebPolynomial functions are functions consisting of numbers and some power of x, e.g. Math is all about solving equations and finding the right answer. It curves back up and passes through (four, zero). So the first thing we need to do is we, Calculate present value of annuity due in excel, Doppler effect enrichment activity answer key, Find the angle between two lines calculator, Find the indicated partial sum calculator, How to do inverse trig functions unit circle, How to find the gradient of a line perpendicular to an equation, How to graph inverse trig functions with transformations. The polynomial function must include all of the factors without any additional unique binomial factors. expression where that is true. So let's look for an A local maximum or local minimum at x= a(sometimes called the relative maximum or minimum, respectively) is the output at the highest or lowest point on the graph in an open interval around x= a. Each x-intercept corresponds to a zero of the polynomial function and each zero yields a factor, so we can now write the polynomial in factored form. Functions can be called all sorts of names. Together, this gives us, [latex]f\left(x\right)=a\left(x+3\right){\left(x - 2\right)}^{2}\left(x - 5\right)[/latex]. Direct link to Darshan's post How can i score an essay , Posted 2 years ago. 1. If you're looking for a punctual person, you can always count on me. 2003-2023 Chegg Inc. All rights reserved. Compare the numbers of bumps If you're seeing this message, it means we're having trouble loading external resources on our website. The remainder = f(a). Direct link to SOULAIMAN986's post In the last question when, Posted 4 years ago. This graph has three x-intercepts: x= 3, 2, and 5. I still don't fully understand how dividing a polynomial expression works. Add comment. Posted 2 years ago. To determine what the math problem is, you will need to take a close look at the information given and use your problem-solving skills. these times constants. WebWrite an equation for the polynomial graphed below Show transcribed image text Expert Answer 100% (3 ratings) From the graph we observe that The zeros of y (x) are x = -4, x = More. Whether you're looking for a new career or simply want to learn from the best, these are the professionals you should be following. The revenue in millions of dollars for a fictional cable company from 2006 through 2013 is shown in the table below. If you're seeing this message, it means we're having trouble loading external resources on our website. Use y for the A simple random sample of 64 households is to be contacted and the sample proportion compu Direct link to Katelyn Clark's post The infinity symbol throw, Posted 5 years ago. An open-top box is to be constructed by cutting out squares from each corner of a 14 cm by 20 cm sheet of plastic then folding up the sides. Because a height of 0 cm is not reasonable, we consider only the zeros 10 and 7. Direct link to Michael Vautier's post The polynomial remainder , Posted 2 years ago. Zero times something, times something is going to be equal to zero. ", To determine the end behavior of a polynomial. Direct link to loumast17's post End behavior is looking a. So pause this video and see Sal said 3/2 instead of 1.5 because 1.5 in fraction form is 3/2. Direct link to Rutwik Pasani's post Why does the graph only t, Posted 7 years ago. The graph curves up from left to right passing through (one, zero). This is a sad thing to say but this is the bwat math teacher I've ever had. Mathematics College answered expert verified Write an equation for the polynomial graphed below 1 See answer Advertisement Advertisement joaobezerra joaobezerra Using the Factor Theorem, the equation for the graphed polynomial is: y(x) = Question: U pone Write an equation for the 4th degree polynomial graphed below. 51 3- 24 1+ -54-32 1 2 345 -2 -3 -4 -5+ y (x)%3D Expert Solution It would be best to put the terms of the polynomial in order from greatest exponent to least exponent before you evaluate the behavior. thanks in advance!! Wolfram alpha free option does not offer as much detail as this one and on top of that I only need to scan the problem with my phone and it breaks it down for me. Write an equation for the polynomial graphed below. For example, consider. Discriptive or inferential, It costs $1,400 to manufacture 100 designer shoes, and $4,100 to manufacture 400 designer shoes. So, to find the polynomial equation we need to, Writing Equations of Polynomial Functions from Graphs. 1. Even then, finding where extrema occur can still be algebraically challenging. Use k if your leading coefficient is positive and-k if your leading coefficlent. WebHow to find 4th degree polynomial equation from given points? WebWrite an equation for the function graphed below Given: The graph of the polynomial is shown below: From the above graph, it can be observed that there are four x x intercepts at x=-3,x=-2,x=1andx=3 x The polynomial remainder theorem states that if any given function f(x) is divided by a polynomial of the form (x - a), f(a) = the remainder of the polynomial division. Learn more about graphed functions here:. A cubic function is graphed on an x y coordinate plane. You have an exponential function. Example: Writing a Formula for a Polynomial Function from Its Graph Write a formula for the polynomial function. 1 has multiplicity 3, and -2 has multiplicity 2. Choose all answers that apply: x+4 x +4 A x+4 x +4 x+3 x +3 B x+3 x +3 x+1 x +1 C x+1 x +1 x x D x x x-1 x 1 E x-1 x 1 x-3 x 3 F x-3 x 3 x-4 x 4 We can estimate the maximum value to be around 340 cubic cm, which occurs when the squares are about 2.75 cm on each side. We now know how to find the end behavior of monomials. Direct link to QUINN767's post It depends on the job tha, Posted 7 years ago. Odd Positive Graph goes down to the far left and up to the far right. So choice D is looking very good. four is equal to zero. Process for Finding Rational ZeroesUse the rational root theorem to list all possible rational zeroes of the polynomial P (x) P ( x).Evaluate the polynomial at the numbers from the first step until we find a zero. Repeat the process using Q(x) Q ( x) this time instead of P (x) P ( x). This repeating will continue until we reach a second degree polynomial. The graph curves up from left to right passing through the negative x-axis side, curving down through the origin, and curving back up through the positive x-axis. b) What percentage of years will have an annual rainfall of more than 38 inches? If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. Now that we know how to find zeros of polynomial functions, we can use them to write formulas based on graphs. x, equals, start color #01a995, k, end color #01a995, f, left parenthesis, x, right parenthesis, equals, 0, start color #01a995, k, end color #01a995, left parenthesis, start color #01a995, k, end color #01a995, comma, 0, right parenthesis, y, equals, f, left parenthesis, x, right parenthesis, x, minus, start color #01a995, k, end color #01a995, f, left parenthesis, x, right parenthesis, g, left parenthesis, x, right parenthesis, equals, left parenthesis, x, minus, 3, right parenthesis, left parenthesis, x, plus, 2, right parenthesis, g, left parenthesis, x, right parenthesis, equals, left parenthesis, x, minus, 3, right parenthesis, left parenthesis, x, minus, left parenthesis, minus, 2, right parenthesis, right parenthesis, g, left parenthesis, x, right parenthesis, left parenthesis, x, minus, start color #01a995, 3, end color #01a995, right parenthesis, left parenthesis, x, minus, left parenthesis, start color #01a995, minus, 2, end color #01a995, right parenthesis, right parenthesis, g, left parenthesis, x, right parenthesis, equals, 0, x, equals, start color #01a995, 3, end color #01a995, x, equals, start color #01a995, minus, 2, end color #01a995, start color #01a995, 3, end color #01a995, start color #01a995, minus, 2, end color #01a995, y, equals, g, left parenthesis, x, right parenthesis, 0, equals, g, left parenthesis, x, right parenthesis, left parenthesis, start color #01a995, 3, end color #01a995, comma, 0, right parenthesis, left parenthesis, start color #01a995, minus, 2, end color #01a995, comma, 0, right parenthesis, f, left parenthesis, x, right parenthesis, equals, left parenthesis, x, plus, 4, right parenthesis, left parenthesis, x, minus, 7, right parenthesis, left parenthesis, minus, 4, comma, 0, right parenthesis, left parenthesis, 7, comma, 0, right parenthesis, left parenthesis, 4, comma, 0, right parenthesis, left parenthesis, minus, 7, comma, 0, right parenthesis, left parenthesis, 2, comma, 0, right parenthesis, 2, slash, 3, space, start text, p, i, end text, h, left parenthesis, x, right parenthesis, h, left parenthesis, x, right parenthesis, equals, left parenthesis, x, minus, 1, right parenthesis, left parenthesis, x, plus, 3, right parenthesis, h, left parenthesis, x, right parenthesis, equals, left parenthesis, x, plus, 1, right parenthesis, left parenthesis, x, minus, 3, right parenthesis, h, left parenthesis, x, right parenthesis, equals, left parenthesis, x, minus, 1, right parenthesis, left parenthesis, x, minus, 3, right parenthesis, h, left parenthesis, x, right parenthesis, equals, left parenthesis, x, plus, 1, right parenthesis, left parenthesis, x, plus, 3, right parenthesis, f, left parenthesis, x, right parenthesis, equals, left parenthesis, x, minus, 1, right parenthesis, left parenthesis, x, minus, 4, right parenthesis, start superscript, start color #aa87ff, 2, end color #aa87ff, end superscript, start color #aa87ff, 2, end color #aa87ff, left parenthesis, x, minus, 4, right parenthesis, f, left parenthesis, x, right parenthesis, equals, left parenthesis, x, minus, 1, right parenthesis, start color #aa87ff, left parenthesis, x, minus, 4, right parenthesis, left parenthesis, x, minus, 4, right parenthesis, end color #aa87ff, f, left parenthesis, x, right parenthesis, equals, left parenthesis, x, minus, 3, right parenthesis, left parenthesis, x, minus, 1, right parenthesis, cubed, g, left parenthesis, x, right parenthesis, equals, left parenthesis, x, plus, 1, right parenthesis, cubed, left parenthesis, 2, x, plus, 1, right parenthesis, squared, minus, start fraction, 1, divided by, 2, end fraction, start fraction, 1, divided by, 2, end fraction, f, left parenthesis, x, right parenthesis, equals, left parenthesis, x, minus, 1, right parenthesis, left parenthesis, x, minus, 4, right parenthesis, squared, g, left parenthesis, x, right parenthesis, equals, left parenthesis, x, minus, 1, right parenthesis, squared, left parenthesis, x, minus, 4, right parenthesis, h, left parenthesis, x, right parenthesis, equals, x, squared, left parenthesis, x, minus, 3, right parenthesis, f, left parenthesis, x, right parenthesis, equals, minus, x, cubed, plus, 4, x, squared, minus, 4, x. A polynomial labeled y equals f of x is graphed on an x y coordinate plane. WebWrite an equation for the polynomial graphed below 4 3 2. For p of three to be equal to zero, we could have an expression like x minus three in the product because this is equal to zero when x is equal to three, and we indeed have that right over there. If a function has a local maximum at a, then [latex]f\left(a\right)\ge f\left(x\right)[/latex] for all xin an open interval around x =a. If a polynomial of lowest degree phas zeros at [latex]x={x}_{1},{x}_{2},\dots ,{x}_{n}[/latex],then the polynomial can be written in the factored form: [latex]f\left(x\right)=a{\left(x-{x}_{1}\right)}^{{p}_{1}}{\left(x-{x}_{2}\right)}^{{p}_{2}}\cdots {\left(x-{x}_{n}\right)}^{{p}_{n}}[/latex]where the powers [latex]{p}_{i}[/latex]on each factor can be determined by the behavior of the graph at the corresponding intercept, and the stretch factor acan be determined given a value of the function other than the x-intercept. Direct link to Judith Gibson's post The question asks about t, Posted 5 years ago. if you can figure that out. Direct link to Lara ALjameel's post Graphs of polynomials eit, Posted 6 years ago. That is what is happening in this equation. Typically when given only zeroes and you want to find the equation through those zeroes, you don't need to worry about the specifics of the graph itself as long as you match it's zeroes. The graph curves down from left to right touching the origin before curving back up. For problem Check Your Understanding 6), if its "6", then why is it odd, not even? Wish it was a tad cheaper but it's the best you can buy for solving math problems of all kinds. It would be best to , Posted a year ago. Question: Write an equation for the 4th degree polynomial graphed below. FYI you do not have a polynomial function. Direct link to s1870299's post how to solve math, Passport to Advanced Math: lessons by skill, f, left parenthesis, x, right parenthesis, equals, x, cubed, plus, 2, x, squared, minus, 5, x, minus, 6, f, left parenthesis, x, right parenthesis, equals, left parenthesis, x, plus, 3, right parenthesis, left parenthesis, x, plus, 1, right parenthesis, left parenthesis, x, minus, 2, right parenthesis, y, equals, left parenthesis, x, minus, start color #7854ab, a, end color #7854ab, right parenthesis, left parenthesis, x, minus, start color #ca337c, b, end color #ca337c, right parenthesis, left parenthesis, x, minus, start color #208170, c, end color #208170, right parenthesis, left parenthesis, start color #7854ab, a, end color #7854ab, comma, 0, right parenthesis, left parenthesis, start color #ca337c, b, end color #ca337c, comma, 0, right parenthesis, left parenthesis, start color #208170, c, end color #208170, comma, 0, right parenthesis, y, equals, left parenthesis, x, plus, 3, right parenthesis, left parenthesis, x, plus, 1, right parenthesis, left parenthesis, x, minus, 2, right parenthesis, start color #7854ab, minus, 3, end color #7854ab, start color #ca337c, minus, 1, end color #ca337c, start color #208170, 2, end color #208170, start color #7854ab, minus, 3, end color #7854ab, plus, 3, equals, 0, start color #ca337c, minus, 1, end color #ca337c, plus, 1, equals, 0, start color #208170, 2, end color #208170, minus, 2, equals, 0, y, equals, left parenthesis, 2, x, minus, 1, right parenthesis, left parenthesis, x, minus, 3, right parenthesis, left parenthesis, x, plus, 5, right parenthesis, p, left parenthesis, x, right parenthesis, y, equals, x, cubed, plus, 2, x, squared, minus, 5, x, minus, 6, start color #7854ab, a, end color #7854ab, x, start superscript, start color #ca337c, n, end color #ca337c, end superscript, start color #7854ab, a, end color #7854ab, is greater than, 0, start color #7854ab, a, end color #7854ab, is less than, 0, start color #ca337c, n, end color #ca337c, start color #7854ab, 1, end color #7854ab, x, start superscript, start color #ca337c, 3, end color #ca337c, end superscript, start color #7854ab, 1, end color #7854ab, is greater than, 0, start color #ca337c, 3, end color #ca337c, f, left parenthesis, x, right parenthesis, equals, minus, 2, x, start superscript, 4, end superscript, minus, 7, x, cubed, plus, 8, x, squared, minus, 10, x, minus, 1, minus, 2, x, start superscript, 4, end superscript, Intro to the Polynomial Remainder Theorem, p, left parenthesis, a, right parenthesis, p, left parenthesis, a, right parenthesis, equals, 0, left parenthesis, a, comma, 0, right parenthesis, p, left parenthesis, a, right parenthesis, does not equal, 0, g, left parenthesis, x, right parenthesis, g, left parenthesis, 0, right parenthesis, equals, minus, 5, g, left parenthesis, 1, right parenthesis, equals, 0, f, left parenthesis, x, right parenthesis, equals, left parenthesis, x, plus, 2, right parenthesis, left parenthesis, x, minus, 2, right parenthesis, left parenthesis, x, minus, 7, right parenthesis, f, left parenthesis, x, right parenthesis, equals, left parenthesis, x, plus, 7, right parenthesis, left parenthesis, x, plus, 2, right parenthesis, left parenthesis, x, minus, 2, right parenthesis, f, left parenthesis, x, right parenthesis, equals, left parenthesis, x, plus, 2, right parenthesis, squared, left parenthesis, x, minus, 7, right parenthesis, f, left parenthesis, x, right parenthesis, equals, left parenthesis, x, minus, 2, right parenthesis, squared, left parenthesis, x, plus, 7, right parenthesis, h, left parenthesis, t, right parenthesis, h, left parenthesis, minus, 1, right parenthesis.
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