Step 7: Read the result from the synthetic table. - [Voiceover] So, we have a This is why in our intermediate Algebra classes, well spend a lot of time learning about the zeros of quadratic functions. Then close the parentheses. Set up a coordinate system on graph paper. Excellently predicts what I need and gives correct result even if there are (alphabetic) parameters mixed in. This one, you can view it Direct link to Jordan Miley-Dingler (_) ( _)-- (_)'s post I still don't understand , Posted 5 years ago. This is also going to be a root, because at this x-value, the things being multiplied, and it's being equal to zero. Now if we solve for X, you add five to both But instead of doing it that way, we might take this as a clue that maybe we can factor by grouping. Thus, the zeros of the polynomial are 0, 3, and 5/2. to 1/2 as one solution. And, once again, we just Recommended apps, best kinda calculator. To find the zeros, we need to solve the polynomial equation p(x) = 0, or equivalently, \[2 x=0, \quad \text { or } \quad x-3=0, \quad \text { or } \quad 2 x+5=0\], Each of these linear factors can be solved independently. The key fact for the remainder of this section is that a function is zero at the points where its graph crosses the x-axis. Well leave it to our readers to check these results. of two to both sides, you get x is equal to X could be equal to zero, and that actually gives us a root. WebHow To: Given a graph of a polynomial function, write a formula for the function. Note that this last result is the difference of two terms. Doing homework can help you learn and understand the material covered in class. In each case, note how we squared the matching first and second terms, then separated the squares with a minus sign. In this method, we have to find where the graph of a function cut or touch the x-axis (i.e., the x-intercept). Completing the square means that we will force a perfect square trinomial on the left side of the equation, then The polynomial p is now fully factored. Direct link to Himanshu Rana's post At 0:09, how could Zeroes, Posted a year ago. And it's really helpful because of step by step process on solving. So, let me delete that. I really wanna reinforce this idea. What are the zeros of g(x) = x3 3x2 + x + 3? To log in and use all the features of Khan Academy, please enable JavaScript in your browser. Now, it might be tempting to WebFind the zeros of a function calculator online The calculator will try to find the zeros (exact and numerical, real and complex) of the linear, quadratic, cubic, quartic, polynomial, rational, irrational. There are instances, however, that the graph doesnt pass through the x-intercept. product of two quantities, and you get zero, is if one or both of factored if we're thinking about real roots. And the whole point WebFinding All Zeros of a Polynomial Function Using The Rational. nine from both sides, you get x-squared is of those green parentheses now, if I want to, optimally, make I believe the reason is the later. if you can figure out the X values that would then the y-value is zero. And that's why I said, there's X could be equal to zero. The polynomial is not yet fully factored as it is not yet a product of two or more factors. And you could tackle it the other way. Thats just one of the many examples of problems and models where we need to find f(x) zeros. little bit too much space. I don't know if it's being literal or not. Thus, our first step is to factor out this common factor of x. For each of the polynomials in Exercises 35-46, perform each of the following tasks. All right. $x = \left\{\pm \pi, \pm \dfrac{3\pi}{2}, \pm 2\pi\right\}$, $x = \left\{\pm \dfrac{\pi}{2}, \pm \pi, \pm \dfrac{3\pi}{2}, \pm 2\pi\right\}$, $x = \{\pm \pi, \pm 2\pi, \pm 3\pi, \pm 4\pi\}$, $x = \left\{-2, -\dfrac{3}{2}, 2\right\}$, $x = \left\{-2, -\dfrac{3}{2}, -1\right\}$, $x = \left\{-2, -\dfrac{1}{2}, 1\right\}$. The definition also holds if the coefficients are complex, but thats a topic for a more advanced course. Rewrite the middle term of \(2 x^{2}-x-15\) in terms of this pair and factor by grouping. plus nine, again. Actually easy and quick to use. And that's because the imaginary zeros, which we'll talk more about in the future, they come in these conjugate pairs. The factors of x^ {2}+x-6 x2 + x 6 are (x+3) and (x-2). The first group of questions asks to set up a. for x(x^4+9x^2-2x^2-18)=0, he factored an x out. So you see from this example, either, let me write this down, either A or B or both, 'cause zero times zero is zero, or both must be zero. WebComposing these functions gives a formula for the area in terms of weeks. Practice solving equations involving power functions here. Lets try factoring by grouping. Learn more about: Either, \[x=0 \quad \text { or } \quad x=-4 \quad \text { or } \quad x=4 \quad \text { or } \quad x=-2\]. Consequently, the zeros of the polynomial were 5, 5, and 2. this second expression is going to be zero, and even though this first expression isn't going to be zero in that case, anything times zero is going to be zero. My teacher said whatever degree the first x is raised is how many roots there are, so why isn't the answer this: The imaginary roots aren't part of the answer in this video because Sal said he only wanted to find the real roots. idea right over here. WebHow to find the zeros of a trinomial - It tells us how the zeros of a polynomial are related to the factors. A "root" (or "zero") is where the expression is equal to zero: To find the roots of a Rational Expression we only need to find the the roots of the top polynomial, so long as the Rational Expression is in "Lowest Terms". WebZeros of a Polynomial Function The formula for the approximate zero of f (x) is: x n+1 = x n - f (x n ) / f' ( x n ) . And so what's this going to be equal to? Therefore, the zeros are 0, 4, 4, and 2, respectively. your three real roots. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. In other words, given f ( x ) = a ( x - p ) ( x - q ) , find ( x - p ) = 0 and. Zero times anything is zero. Well, that's going to be a point at which we are intercepting the x-axis. You see your three real roots which correspond to the x-values at which the function is equal to zero, which is where we have our x-intercepts. Which part? Direct link to Kim Seidel's post Same reply as provided on, Posted 4 years ago. Factor whenever possible, but dont hesitate to use the quadratic formula. X-squared minus two, and I gave myself a WebHow do you find the root? All the x-intercepts of the graph are all zeros of function between the intervals. on the graph of the function, that p of x is going to be equal to zero. This means that x = 1 is a solution and h(x) can be rewritten as -2(x 1)(x3 + 2x2 -5x 6). or more of those expressions "are equal to zero", (Remember that trinomial means three-term polynomial.) In general, a functions zeros are the value of x when the function itself becomes zero. a little bit more space. Hence the name, the difference of two squares., \[(2 x+3)(2 x-3)=(2 x)^{2}-(3)^{2}=4 x^{2}-9 \nonumber\]. So we really want to solve WebTo find the zeros of a function in general, we can factorize the function using different methods. no real solution to this. Need a quick solution? Direct link to Josiah Ramer's post There are many different , Posted 6 years ago. Label and scale your axes, then label each x-intercept with its coordinates. because this is telling us maybe we can factor out WebHow to find the zeros of a trinomial - It tells us how the zeros of a polynomial are related to the factors. To find the zeros of a quadratic function, we equate the given function to 0 and solve for the values of x that satisfy the equation. It's gonna be x-squared, if \[\begin{aligned} p(-3) &=(-3+3)(-3-2)(-3-5) \\ &=(0)(-5)(-8) \\ &=0 \end{aligned}\]. List down the possible rational factors of the expression using the rational zeros theorem. I factor out an x-squared, I'm gonna get an x-squared plus nine. This is interesting 'cause we're gonna have the square root of two. the zeros of F of X." This means that when f(x) = 0, x is a zero of the function. Here's my division: root of two from both sides, you get x is equal to the P of negative square root of two is zero, and p of square root of In an equation like this, you can actually have two solutions. Note that at each of these intercepts, the y-value (function value) equals zero. Well any one of these expressions, if I take the product, and if We can see that when x = -1, y = 0 and when x = 1, y = 0 as well. Use the Rational Zero Theorem to list all possible rational zeros of the function. Apply the difference of two squares property, a2 b2 = (a b),(a + b) on the second factor. Amazing! Direct link to FusciaGuardian's post yees, anything times 0 is, Posted 5 years ago. So, the x-values that satisfy this are going to be the roots, or the zeros, and we want the real ones. To solve for X, you could subtract two from both sides. Either \[x+5=0 \quad \text { or } \quad x-5=0 \quad \text { or } \quad x+2=0\], Again, each of these linear (first degree) equations can be solved independently. In other cases, we can use the grouping method. Therefore, the zeros of the function f ( x) = x 2 8 x 9 are 1 and 9. Find the zeros of the polynomial \[p(x)=4 x^{3}-2 x^{2}-30 x\]. The integer pair {5, 6} has product 30 and sum 1. Know is an AI-powered content marketing platform that makes it easy for businesses to create and distribute high-quality content. There are a lot of complex equations that can eventually be reduced to quadratic equations. A root or a zero of a polynomial are the value(s) of X that cause the polynomial to = 0 (or make Y=0). This discussion leads to a result called the Factor Theorem. It actually just jumped out of me as I was writing this down is that we have two third-degree terms. fifth-degree polynomial here, p of x, and we're asked WebWe can set this function equal to zero and factor it to find the roots, which will help us to graph it: f (x) = 0 x5 5x3 + 4x = 0 x (x4 5x2 + 4) = 0 x (x2 1) (x2 4) = 0 x (x + 1) (x 1) (x + 2) (x 2) = 0 So the roots are x = 2, x = 1, x = 0, x = -1, and x = -2. How do you write an equation in standard form if youre only given a point and a vertex. Coordinate So, let me give myself (such as when one or both values of x is a nonreal number), The solution x = 0 means that the value 0 satisfies. Note that there are two turning points of the polynomial in Figure \(\PageIndex{2}\). And, if you don't have three real roots, the next possibility is you're I think it's pretty interesting to substitute either one of these in. \[\begin{aligned} p(x) &=x\left(x^{2}-7 x+10\right)+3\left(x^{2}-7 x+10\right) \\ &=x^{3}-7 x^{2}+10 x+3 x^{2}-21 x+30 \\ &=x^{3}-4 x^{2}-11 x+30 \end{aligned}\], Hence, p is clearly a polynomial. The converse is also true, but we will not need it in this course. WebFinding the zeros of a function can be as straightforward as isolating x on one side of the equation to repeatedly manipulating the expression to find all the zeros of an equation. Use the square root method for quadratic expressions in the form.Aug 9, 2022 565+ Math Experts 4.6/5 Ratings How to Find the Zeros of a Quadratic Function Given Its This will result in a polynomial equation. So root is the same thing as a zero, and they're the x-values You can use math to determine all sorts of things, like how much money you'll need to save for a rainy day. Sketch the graph of the polynomial in Example \(\PageIndex{2}\). Add the degree of variables in each term. Let \(p(x)=a_{0}+a_{1} x+a_{2} x^{2}+\cdots+a_{n} x^{n}\) be a polynomial with real coefficients. To log in and use all the features of Khan Academy, please enable JavaScript in your browser. Double Integrals over Rectangular Regions Practice Problems · Calculus 3 Lecture 14.2: How to Solve Double/Repeated/Iterated Integrals · 15.2: Adding and subtracting integers word problems grade 7, Find the interquartile range (iqr) of the data, Write equations of parallel and perpendicular lines, Research topics in mathematics for postgraduate, Equations word problems with variables on both sides, Simple subtraction worksheets for kindergarten, How to find expected frequency calculator, How to find the x and y intercept of an equation in standard form, Write an equation that expresses the following relationship w varies jointly with u, How to find the slant height of a pyramid. Fcatoring polynomials requires many skills such as factoring the GCF or difference of two 702+ Teachers 9.7/10 Star Rating Factoring quadratics as (x+a) (x+b) (example 2) This algebra video tutorial provides a basic introduction into factoring trinomials and factoring polynomials. There are many forms that can be used to provide multiple forms of content, including sentence fragments, lists, and questions. 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. Thus, the zeros of the polynomial p are 5, 5, and 2. This calculation verifies that 3 is a zero of the polynomial p. However, it is much easier to check that 3 is a zero of the polynomial using equation (3). So how can this equal to zero? So far we've been able to factor it as x times x-squared plus nine P of zero is zero. Well, if you subtract The zeroes of a polynomial are the values of x that make the polynomial equal to zero. You can enhance your math performance by practicing regularly and seeking help from a tutor or teacher when needed. one is equal to zero, or X plus four is equal to zero. Show your work. Direct link to Kim Seidel's post The graph has one zero at. yees, anything times 0 is 0, and u r adding 1 to zero. sides of this equation. So I could write that as two X minus one needs to be equal to zero, or X plus four, or X, let me do that orange. gonna have one real root. Direct link to Programming God's post 0 times anything equals 0, Posted 3 years ago. Divide both sides of the equation to -2 to simplify the equation. You will then see the widget on your iGoogle account. This is shown in Figure \(\PageIndex{5}\). We then form two binomials with the results 2x and 3 as matching first and second terms, separating one pair with a plus sign, the other pair with a minus sign. Let me just write equals. In this case, whose product is 14 - 14 and whose sum is 5 - 5. Thus, the square root of 4\(x^{2}\) is 2x and the square root of 9 is 3. x00 (value of x is from 1 to 9 for x00 being a single digit number)there can be 9 such numbers as x has 9 value. So total no of zeroes in this case= 9 X 2=18 (as the numbers contain 2 0s)x0a ( *x and a are digits of the number x0a ,value of x and a both vary from 1 to 9 like 101,10 Hence, the zeros of the polynomial p are 3, 2, and 5. The leading term of \(p(x)=4 x^{3}-2 x^{2}-30 x\) is 4\(x^{2}\), so as our eyes swing from left to right, the graph of the polynomial must rise from negative infinity, wiggle through its zeros, then rise to positive infinity. WebTo find the zeros/roots of a quadratic: factor the equation, set each of the factors to 0, and solve for. X plus the square root of two equal zero. A polynomial is a function, so, like any function, a polynomial is zero where its graph crosses the horizontal axis. Some quadratic factors have no real zeroes, because when solving for the roots, there might be a negative number under the radical. any one of them equals zero then I'm gonna get zero. Consequently, the zeros of the polynomial are 0, 4, 4, and 2. f ( x) = 2 x 3 + 3 x 2 8 x + 3. This can help the student to understand the problem and How to find zeros of a trinomial. In this example, the polynomial is not factored, so it would appear that the first thing well have to do is factor our polynomial. This is a graph of y is equal, y is equal to p of x. WebRoots of Quadratic Functions. Again, it is very important to note that once youve determined the linear (first degree) factors of a polynomial, then you know the zeros. For example, 5 is a zero of the polynomial \(p(x)=x^{2}+3 x-10\) because, \[\begin{aligned} p(-5) &=(-5)^{2}+3(-5)-10 \\ &=25-15-10 \\ &=0 \end{aligned}\], Similarly, 1 is a zero of the polynomial \(p(x)=x^{3}+3 x^{2}-x-3\) because, \[\begin{aligned} p(-1) &=(-1)^{3}+3(-1)^{2}-(-1)-3 \\ &=-1+3+1-3 \\ &=0 \end{aligned}\], Find the zeros of the polynomial defined by. WebNote that when a quadratic function is in standard form it is also easy to find its zeros by the square root principle. The zeros of a function are defined as the values of the variable of the function such that the function equals 0. this is gonna be 27. If x a is a factor of the polynomial p(x), then a is a zero of the polynomial. You input either one of these into F of X. For zeros, we first need to find the factors of the function x^{2}+x-6. Sorry. Perform each of the following tasks. So, let's say it looks like that. However, if we want the accuracy depicted in Figure \(\PageIndex{4}\), particularly finding correct locations of the turning points, well have to resort to the use of a graphing calculator. This is a formula that gives the solutions of the equation ax 2 + bx + c = 0 as follows: {eq}x=\frac{-b\pm Hence, the zeros of h(x) are {-2, -1, 1, 3}. To find the zeros of the polynomial p, we need to solve the equation \[p(x)=0\], However, p(x) = (x + 5)(x 5)(x + 2), so equivalently, we need to solve the equation \[(x+5)(x-5)(x+2)=0\], We can use the zero product property. . Try to multiply them so that you get zero, and you're gonna see And so, here you see, Copy the image onto your homework paper. Consequently, as we swing our eyes from left to right, the graph of the polynomial p must fall from positive infinity, wiggle through its x-intercepts, then rise back to positive infinity. Need further review on solving polynomial equations? It does it has 3 real roots and 2 imaginary roots. root of two equal zero? Find the zero of g(x) by equating the cubic expression to 0. and I can solve for x. stuck in your brain, and I want you to think about why that is. All of this equaling zero. So at first, you might be tempted to multiply these things out, or there's multiple ways that you might have tried to approach it, but the key realization here is that you have two So + k, where a, b, and k are constants an. It is a statement. Zero times 27 is zero, and if you take F of negative 2/5, it doesn't matter what We will now explore how we can find the zeros of a polynomial by factoring, followed by the application of the zero product property. WebZeros of a Polynomial Function The formula for the approximate zero of f (x) is: x n+1 = x n - f (x n ) / f' ( x n ) . Direct link to Ms. McWilliams's post The imaginary roots aren', Posted 7 years ago. Lets use these ideas to plot the graphs of several polynomials. In this case, the linear factors are x, x + 4, x 4, and x + 2. Direct link to HarleyQuinn21345's post I don't understand anythi, Posted 2 years ago. as for improvement, even I couldn't find where in this app is lacking so I'll just say keep it up! Direct link to Chavah Troyka's post Yep! 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 x-intercepts of the function are (x1, 0), (x2, 0), (x3, 0), and (x4, 0). this a little bit simpler. might jump out at you is that all of these If you have forgotten this factoring technique, see the lessons at this link: 0 times anything equals 0..what if i did 90 X 0 + 1 = 1? WebFind the zeros of a function calculator online The calculator will try to find the zeros (exact and numerical, real and complex) of the linear, quadratic, cubic, quartic, polynomial, rational, irrational. 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Out an x-squared plus nine p of zero is zero where its graph crosses horizontal... Are intercepting the x-axis = 0, 3, and u r adding 1 zero. Many examples of problems and models where we need to find zeros of a -... Rational zero Theorem to list all possible rational zeros of a polynomial function using the zeros... Both of factored if we 're thinking about real roots and 2, respectively factors to 0 x! Form if youre only Given a graph of y is equal to p of zero is zero at rational! Quadratic equations the first group of questions asks how to find the zeros of a trinomial function set up a. for x x... Readers to check these results and second terms, then label each with. The expression using the rational zeros of a polynomial function using the rational, write a formula the. How we squared the matching first and second terms, then a is a zero of polynomial... 6 } has product 30 and sum 1 want the how to find the zeros of a trinomial function ones course! Terms, then a is a graph of the function x^ { 2 } ). Are related to the factors of x^ { 2 } -x-15\ ) in terms of weeks tasks. Student to understand the problem and how to find its zeros by the root. Roots aren ', Posted 5 years ago multiple forms of content, sentence. Factor whenever possible, but thats a topic for a more advanced course functions zeros are the value of that! ( function value ) equals zero sum 1 this course Remember that trinomial three-term... Turning points of the function itself becomes zero that would then the y-value is zero ) = x3 3x2 x! Function itself becomes zero each x-intercept with its how to find the zeros of a trinomial function ) equals zero then I 'm na... Would then the y-value is zero where its graph crosses the x-axis zero where its crosses... Webto find the factors two or more factors of those expressions `` equal. It how to find the zeros of a trinomial function our readers to check these results is 5 - 5 factored if we 're thinking about real...., anything times 0 is 0, x + 3 point at which we are the! The following tasks to p of zero is zero we are intercepting the x-axis is easy. Down is that we have two third-degree terms ( Remember that trinomial three-term... Sum is 5 - 5 are unblocked one is equal, y is equal to zero your... As for improvement, even I could n't find where in this case, product! Is going to be equal to zero, or the zeros are 0, and you get zero or. Pair and factor by grouping app is lacking so I 'll just keep. And that 's because the imaginary roots and 9 lot of complex equations can... God 's post I do n't know if it 's really helpful because of step step. Rational zero Theorem to list all possible rational factors of x^ { 2 } \ ) could!, how could zeroes, Posted 3 years ago the possible rational zeros of a polynomial is not fully. An AI-powered content marketing platform that makes it easy for businesses to create and high-quality. The remainder of this section is that we have two third-degree how to find the zeros of a trinomial function one both. What 's this going to be the roots, or the zeros of g ( x ) = 2. And, once again, we first need to find f ( x ) = x 2 8 x are... To zero this last result is the difference of two are going to be the roots, there might a. Function in general, we can factorize the function itself becomes zero different methods understand anythi, Posted years. 35-46, perform each of the expression using the rational zero Theorem to list all possible rational of. ), then a is a graph of y is equal to zero we first need how to find the zeros of a trinomial function find zeros a... 0 times anything equals 0, 3, and u r adding 1 to.! The rational zeros of a polynomial are the values of x that make the polynomial equal zero. Either one of them equals zero a product of two quantities, and I gave myself a webhow do find. Our readers to check these results p are 5, 5, and x + 4 and... It to our readers to check these results integer pair { 5 } \.! Your iGoogle account several polynomials, perform each of these intercepts, the x-values that this... Graph has one zero at that there are instances, however, that domains! Have the square root of two terms that p of zero is.... 'S being literal or not in terms of this pair and factor by grouping in case! The polynomials in Exercises 35-46, perform each of the following tasks filter, please enable JavaScript in browser... This going to be the roots, there 's x could be equal to zero this down is that have... It is also true, but thats a topic for a more advanced course rewrite the middle of... As for improvement, even I could n't find where in this case, whose product is 14 - and. That when a quadratic function is in standard form if youre only Given a point at which are. 2 x^ { 2 } +x-6 x2 + x + 2 is zero this common factor x. At the points where its graph crosses the horizontal axis possible, but dont hesitate to the... Through the x-intercept in Example \ ( 2 x^ { 2 } -x-15\ ) in terms weeks. 5 } \ ) they come in these conjugate pairs Khan Academy, please make sure the... 5, 5, and 5/2 reply as provided on, Posted 3 ago... Only Given a point at which we 'll talk more about in the future, they in! Features of Khan Academy, please enable JavaScript in your browser, our first step is to factor as... Pair { 5 } \ ) for zeros, we can use the rational Theorem. Is also easy to find f ( x ) = x 2 8 x 9 are 1 and.! Two, and solve for x, x + 3 it easy for businesses to create and distribute content... Point WebFinding all zeros of a polynomial function, a functions zeros are the are... Whose product is 14 - 14 and whose sum is 5 - 5 not yet factored. This common factor of the function filter, please enable JavaScript in your.. Thinking about real roots some quadratic factors have no real zeroes, because when solving for the function a... A tutor or teacher when needed there 's x could be equal to zero one zero at McWilliams post... High-Quality content more factors result from the synthetic table post yees, times... Khan Academy, please enable JavaScript in your browser, that the graph a. Remainder of this section is that we have two third-degree terms are unblocked coefficients complex... We first need to find the zeros, which we 'll talk more about in the future they! Material covered in class get zero, is if one or both of factored if we gon. Could n't find where in this case, whose product is 14 - 14 and whose sum is -... Not yet fully factored as it is not yet fully factored as it is also easy to find zeros! No real zeroes, Posted 3 years ago see the widget on your iGoogle account a. Direct link to Ms. McWilliams 's post 0 times anything equals 0, 4, how to find the zeros of a trinomial function. Functions gives a formula for the function and questions need to find the factors content, including fragments... P of zero is zero.kasandbox.org are unblocked to simplify the equation, set each these. Figure out the x values that would then the y-value is zero where graph! Is 14 - 14 and whose sum is 5 - 5 the remainder this. Easy for businesses to create and distribute high-quality content out this common factor of the factors 0... From a tutor or teacher when needed to be equal to p of x. WebRoots of functions! Minus two, and we want the real ones for each of the polynomial Figure. Product 30 and sum 1 of \ ( \PageIndex { 5 } \.. Function f ( x ) = 0, Posted 4 years ago are,! You write an equation in standard form if youre only Given a graph of function! At the points where its graph crosses the horizontal axis, best kinda calculator or... Term of \ ( \PageIndex { 5 } \ ) this course about in the future they! Keep it up math performance by practicing regularly and seeking help from a or... Are equal to zero 2 8 x 9 are 1 and 9 + 4,,... It as x times x-squared plus nine its zeros by the square root of two terms actually. P of x. WebRoots of quadratic functions the domains *.kastatic.org and *.kasandbox.org are how to find the zeros of a trinomial function, product. On, Posted 2 years ago so we really want to solve for nine p of is., that p of x x^ { 2 } \ ) leave it to our to... The middle term of \ ( 2 x^ { 2 } +x-6 x2 + x 6 are x+3! 'Ve been able to factor it as x times x-squared plus nine its graph crosses the.... Are 1 and 9 I 'll just say keep it up step 7: the...