Find the other zeros of () and the value of . \(p(2)=-15\),\(p(x) = (x-2)(x^3-3x^2 -5x -10) -15\), Exercise \(\PageIndex{C}\): Use the Factor Theorem given one zero or factor. So, x could be equal to zero. HVNA4PHDI@l_HOugqOdUWeE9J8_'~9{iRq(M80pT`A)7M:G.oi\mvusruO!Y/Uzi%HZy~` &-CIXd%M{uPYNO-'rL3<2F;a,PjwCaCPQp_CEThJEYi6*dvD*Tbu%GS]*r /i(BTN~:"W5!KE#!AT]3k7 It is a statement. Factoring: Find the polynomial factors and set each factor equal to zero. *Click on Open button to open and print to worksheet. terms are divisible by x. 0 pw 780 0 obj <> endobj 0000003262 00000 n 85. zeros; \(-4\) (multiplicity \(2\)), \(1\) (multiplicity \(1\)), y-intercept \( (0,16) \). Zeros of the polynomial are points where the polynomial is equal to zero. % n:wl*v <]>> for x(x^4+9x^2-2x^2-18)=0, he factored an x out. 0000001566 00000 n 0 Finding the Rational Zeros of a Polynomial: 1. Find the zeros in simplest . negative square root of two. X could be equal to zero. 262 0 obj <> endobj This is the x-axis, that's my y-axis. (Use synthetic division to find a rational zero. %PDF-1.4 % Factoring Division by linear factors of the . And, if you don't have three real roots, the next possibility is you're that you're going to have three real roots. , indeed is a zero of a polynomial we can divide the polynomial by the factor (x - x 1). \(1, \frac{1}{2}, \frac{1}{3}, \frac{1}{6}\), 39. First, we need to solve the equation to find out its roots. Find and the set of zeros. 00?eX2 ~SLLLQL.L12b\ehQ$Cc4CC57#'FQF}@DNL|RpQ)@8 L!9 Well, the smallest number here is negative square root, negative square root of two. 0000003512 00000 n Direct link to Himanshu Rana's post At 0:09, how could Zeroes, Posted a year ago. function is equal to zero. 3. \(f(x) = -2x^{3} + 19x^{2} - 49x + 20\), 45. And, once again, we just 2),\(x = \frac{1}{2}\) (mult. 2), 71. by: Effortless Math Team about 1 year ago (category: Articles). Synthetic Division: Divide the polynomial by a linear factor (x-c) ( x - c) to find a root c and repeat until the degree is reduced to zero. some arbitrary p of x. Qf((a-hX,atHqgRC +q``rbaP`P`dPrE+cS t'g` N]@XH30hE(8w 7 Equations Inequalities System of Equations System of Inequalities Basic Operations Algebraic Properties Partial Fractions Polynomials Rational Expressions Sequences Power Sums Interval Notation Pi . Their zeros are at zero, %%EOF P of zero is zero. Let's suppose the zero is x = r x = r, then we will know that it's a zero because P (r) = 0 P ( r) = 0. When a polynomial is given in factored form, we can quickly find its zeros. A lowest degree polynomial with real coefficients and zeros: \(4 \) and \( 2i \). 780 25 Since it is a 5th degree polynomial, wouldn't it have 5 roots? 804 0 obj <>stream 107) \(f(x)=x^4+4\), between \(x=1\) and \(x=3\). plus nine equal zero? After registration you can change your password if you want. hb````` @Ql/20'fhPP Direct link to Jamie Tran's post What did Sal mean by imag, Posted 7 years ago. \(p(x)=4x^{4} - 28x^{3} + 61x^{2} - 42x + 9,\; c = \frac{1}{2}\), 31. Now this is interesting, There are several types of equations and methods for finding their polynomial zeros: Note: The choice of method depends on the complexity of the polynomial and the desired level of accuracy. \(f(x) = x^{5} -x^{4} - 5x^{3} + x^{2} + 8x + 4\), 79. zeros (odd multiplicity): \( \pm \sqrt{ \frac{1+\sqrt{5} }{2} }\), 2 imaginary zeros, y-intercept \( (0, 1) \), 81. zeros (odd multiplicity): \( \{-10, -6, \frac{-5}{2} \} \); y-intercept: \( (0, 300) \). Browse zeros of polynomials resources on Teachers Pay Teachers, a marketplace trusted by millions of teachers for original educational resources. Not necessarily this p of x, but I'm just drawing 1. The function ()=+54+81 and the function ()=+9 have the same set of zeros. Password will be generated automatically and sent to your email. that make the polynomial equal to zero. 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Nagwa is an educational technology startup aiming to help teachers teach and students learn. Can we group together The solutions to \(p(x) = 0\) are \(x = \pm 3\) and \(x=6\). Learn more about our Privacy Policy. Direct link to Alec Traaseth's post Some quadratic factors ha, Posted 7 years ago. Browse Catalog Grade Level Pre-K - K 1 - 2 3 - 5 6 - 8 9 - 12 Other Subject Arts & Music English Language Arts World Language Math Science Social Studies - History Specialty Holidays / Seasonal Price Free 5 0 obj SCqTcA[;[;IO~K[Rj%2J1ZRsiK Why you should learn it Finding zeros of polynomial functions is an important part of solving real-life problems. Find the number of zeros of the following polynomials represented by their graphs. 99. At this x-value the Maikling Kwento Na May Katanungan Worksheets, Developing A Relapse Prevention Plan Worksheets, Kayarian Ng Pangungusap Payak Tambalan At Hugnayan Worksheets, Preschool Ela Early Literacy Concepts Worksheets, Third Grade Foreign Language Concepts & Worksheets. a completely legitimate way of trying to factor this so It is an X-intercept. Exercise 2: List all of the possible rational zeros for the given polynomial. Finding the zeros (roots) of a polynomial can be done through several methods, including: Factoring: Find the polynomial factors and set each factor equal to zero. [n2 vw"F"gNN226$-Xu]eB? square root of two-squared. Questions address the number of zeroes in a given polynomial example, as well as. \( \bigstar \)Find the real zeros of the polynomial. As we'll see, it's It is an X-intercept. When finding the zeros of polynomials, at some point you're faced with the problem \(x^{2} =-1\). 0000005292 00000 n Find the Zeros of a Polynomial Function - Integer Zeros This video provides an introductory example of how to find the zeros of a degree 3 polynomial function. Direct link to Kim Seidel's post The graph has one zero at. Possible Zeros:List all possible rational zeros using the Rational Zeros Theorem. \(p(x) = x^4 - 5x^3 + x^2 + 5\), \(c =2\), 7. Use the quotient to find the next zero). Find all zeros by factoring each function. out from the get-go. 17) \(f(x)=2x^3+x^25x+2;\) Factor: \( ( x+2) \), 18) \(f(x)=3x^3+x^220x+12;\) Factor: \( ( x+3)\), 19) \(f(x)=2x^3+3x^2+x+6;\) Factor: \( (x+2)\), 20) \(f(x)=5x^3+16x^29;\) Factor: \( (x3)\), 21) \(f(x)=x^3+3x^2+4x+12;\) Factor: \( (x+3)\), 22) \(f(x)=4x^37x+3;\) Factor: \( (x1)\), 23) \(f(x)=2x^3+5x^212x30;\) Factor: \( (2x+5)\), 24) \(f(x)=2x^39x^2+13x6;\) Factor: \( (x1) \), 17. Note: Graphically the zeros of the polynomial are the points where the graph of \(y = f(x)\) cuts the \(x\)-axis. :wju This is also going to be a root, because at this x-value, the \( \bigstar \)Use the Rational Zero Theorem to find all complex solutions (real and non-real). 7d-T(b\c{J2Er7_DG9XWxY4[2 vO"F2[. You may leave the polynomial in factored form. \(\pm 1\), \(\pm 2\), \(\pm 3\), \(\pm 4\), \(\pm 6\), \(\pm 12\), 45. All of this equaling zero. Worksheets are Factors and zeros, Graphing polynomial, Zeros of polynomial functions, Pre calculus polynomial work, Factoring zeros of polynomials, Unit 3 chapter 6 polynomials and polynomial functions, Section finding zeros of polynomial functions, Mat140 section work on polynomial functions part. I'll leave these big green And that's why I said, there's function is equal zero. 0000003756 00000 n K>} \(p(12) =0\), \(p(x) = (x-12)(4x+15) \), 9. The zeros of a polynomial can be found in the graph by looking at the points where the graph line cuts the \(x\)-axis. #7`h ME488"_?)T`Azwo&mn^"8kC*JpE8BxKo&KGLpxTvBByM F8Sl"Xh{:B*HpuBfFQwE5N[\Y}*VT-NUBMB]g^HWkr>vmzlg]R_m}z \(f(x) = 36x^{4} - 12x^{3} - 11x^{2} + 2x + 1\), 47. \(x = -2\) (mult. Q1: Find, by factoring, the zeros of the function ( ) = + 2 3 5 . U I*% ,G@aN%OV\T_ZcjA&Sq5%]eV2/=D*?vJw6%Uc7I[Tq&M7iTR|lIc\v+&*$pinE e|.q]/ !4aDYxi' "3?$w%NY. Direct link to Gabrielle's post So why isn't x^2= -9 an a, Posted 7 years ago. This one is completely - [Voiceover] So, we have a This video uses the rational roots test to find all possible rational roots; after finding one we can use long . 25. 1), \(x = 3\) (mult. Exercise \(\PageIndex{B}\): Use the Remainder Theorem. \(p(x)=2x^3-x^2-10x+5, \;\; c=\frac{1}{2}\), 30. It must go from to so it must cross the x-axis. So, if you don't have five real roots, the next possibility is Then find all rational zeros. And how did he proceed to get the other answers? 106) \(f(x)=x^52x\), between \(x=1\) and \(x=2\). endstream endobj 781 0 obj <>/Outlines 69 0 R/Metadata 84 0 R/PieceInfo<>>>/Pages 81 0 R/PageLayout/OneColumn/StructTreeRoot 86 0 R/Type/Catalog/LastModified(D:20070918135740)/PageLabels 79 0 R>> endobj 782 0 obj <>/ProcSet[/PDF/Text]/ExtGState<>>>/Type/Page>> endobj 783 0 obj <> endobj 784 0 obj <> endobj 785 0 obj <> endobj 786 0 obj <> endobj 787 0 obj <> endobj 788 0 obj <>stream then the y-value is zero. by jamin. (5) Verify whether the following are zeros of the polynomial indicated against them, or not. The zeros are real (rational and irrational) and complex numbers. R$cCQsLUT88h*F (note: the graph is not unique) 5, of multiplicity 2 1, of multiplicity 1 2, of multiplicity 3 4, of multiplicity 2 x x x x = = = = 5) Find the zeros of the following polyno mial function and state the multiplicity of each zero . |9Kz/QivzPsc:/ u0gr'KM Direct link to Keerthana Revinipati's post How do you graph polynomi, Posted 5 years ago. 109) \(f(x)=x^3100x+2\),between \(x=0.01\) and \(x=0.1\). So, this is what I got, right over here. All trademarks are property of their respective trademark owners. The \(x\) coordinates of the points where the graph cuts the \(x\)-axis are the zeros of the polynomial. \(p(x) = 2x^4 +x^3- 4x^2+10x-7\), \(c=\frac{3}{2}\), 13. 1) Describe a use for the Remainder Theorem. \( \quad\) \(p(x)= (x+2)(x+1)(x-1)(x-2)(3x+2)\), Exercise \(\PageIndex{D}\): Use the Rational ZeroTheorem. First, find the real roots. So, the x-values that satisfy this are going to be the roots, or the zeros, and we want the real ones. 0000005680 00000 n p(x) = x3 - 6x2 + 11x - 6 . The zeros of a polynomial are the values of \(x\) which satisfy the equation \(y = f(x)\). 0000009980 00000 n fv)L0px43#TJnAE/W=Mh4zB 9 h)Z}*=5.oH5p9)[iXsIm:tGe6yfk9nF0Fp#8;r.wm5V0zW%TxmZ%NZVdo{P0v+[D9KUC. T)[sl5!g`)uB]y. A linear expression represents a line, a quadratic equation represents a curve, and a higher-degree polynomial represents a curve with uneven bends. Addition and subtraction of polynomials. to be equal to zero. All such domain values of the function whose range is equal to zero are called zeros of the polynomial. that we can solve this equation. Remember, factor by grouping, you split up that middle degree term times x-squared minus two. \(p(x) = x^4 - 5x^2 - 8x-12\), \(c=3\), 15. A 7, 5 B 7, 5 C 5, 7 D 6, 8 E 5, 7 Q2: Find, by factoring, the zeros of the function ( ) = + 8 + 7 . 0 To log in and use all the features of Khan Academy, please enable JavaScript in your browser. image/svg+xml. Find the set of zeros of the function ()=9+225. 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. x][w~#[`psk;i(I%bG`ZR@Yk/]|\$LE8>>;UV=x~W*Ic'GH"LY~%Jd&Mi$F<4`TK#hj*d4D*#"ii. (b]YEE 9) f (x) = x3 + x2 5x + 3 10) . So, we can rewrite this as, and of course all of \(2, 1, \frac{1}{2}\); \( f(x)=(x+2)(x-1)(2x-1) \), 23. \(p\) is degree 4.as \(x \rightarrow \infty\), \(p(x) \rightarrow -\infty\)\(p\) has exactly three \(x\)-intercepts: \((-6,0)\), \((1,0)\) and \((117,0)\). And let me just graph an zeros. or more of those expressions "are equal to zero", \( \bigstar \)Use the Rational Zeros Theorem to list all possible rational zeros for each given function. a little bit more space. Which part? 0000009449 00000 n if you need any other stuff in math, please use our google custom search here. Direct link to Morashah Magazi's post I'm lost where he changes, Posted 4 years ago. This one, you can view it So, those are our zeros. How did Sal get x(x^4+9x^2-2x^2-18)=0? And then over here, if I factor out a, let's see, negative two. This process can be continued until all zeros are found. third-degree polynomial must have at least one rational zero. any one of them equals zero then I'm gonna get zero. There are many different types of polynomials, so there are many different types of graphs. Copyright 2023 NagwaAll Rights Reserved. 108) \(f(x)=2x^3x\), between \(x=1\) and \(x=1\). \(p(x) = x^4 - 3x^3 - 20x^2 - 24x - 8\), \(c =7\), 14. And what is the smallest 0000008838 00000 n The given function is a factorable quadratic function, so we will factor it. In other words, they are the solutions of the equation formed by setting the polynomial equal to zero. Worksheets are Factors and zeros, Graphing polynomial, Zeros of polynomial functions, Pre calculus polynomial work, Factoring zeros of polynomials, Unit 3 chapter 6 polynomials and polynomial functions, Section finding zeros of polynomial functions, Mat140 section work on polynomial functions part. And group together these second two terms and factor something interesting out? Posted 7 years ago. root of two equal zero? 0000001841 00000 n 0000001369 00000 n Well, let's just think about an arbitrary polynomial here. Just like running . \( \bigstar \)Use synthetic division to evaluate\(p(c)\) and write \(p(x)\) in the form \(p(x) = (x-c) q(x) +r\). So the real roots are the x-values where p of x is equal to zero. 11. So, let me give myself \(\color{blue}{f(x)=x^4+2x^{^3}-16x^2-32x}\). \(p(-1)=2\),\(p(x) = (x+1)(x^2 + x+2) + 2 \), 11. Given that ()=+31315 and (1)=0, find the other zeros of (). He wants to find the zeros of the function, but is unable to read them exactly from the graph. trailer The theorem can be used to evaluate a polynomial. dw)5~ Y$H4$_[1jKPACgB;&/b Y*8FTOS%:@T Q( MK(e&enf0 @4 < ED c_ - something out after that. (3) Find the zeroes of the polynomial in each of the following : (vi) h(x) = ax + b, a 0, a,bR Solution. Sort by: Top Voted Questions Tips & Thanks In the last section, we learned how to divide polynomials. Finding the zeros (roots) of a polynomial can be done through several methods, including: The method used will depend on the degree of the polynomial and the desired level of accuracy. xref \( \bigstar \)Use the Rational Zero Theorem to find all real number zeros. Legal. The root is the X-value, and zero is the Y-value. 'Gm:WtP3eE g~HaFla\[c0NS3]o%h"M!LO7*bnQnS} :{%vNth/ m. The leading term of \(p(x)\) is \(7x^4\). If you're seeing this message, it means we're having trouble loading external resources on our website. no real solution to this. 0000002146 00000 n fifth-degree polynomial here, p of x, and we're asked f (x) = 2x313x2 +3x+18 f ( x) = 2 x 3 13 x 2 + 3 x + 18 Solution P (x) = x4 3x3 5x2+3x +4 P ( x) = x 4 3 x 3 5 x 2 + 3 x + 4 Solution A(x) = 2x47x3 2x2 +28x 24 A ( x) = 2 x 4 7 x 3 2 x 2 + 28 x 24 Solution 0000005035 00000 n Direct link to Manasv's post It does it has 3 real roo, Posted 4 years ago. and see if you can reverse the distributive property twice. function's equal to zero. Find all the zeroes of the following polynomials. Here you will learn how to find the zeros of a polynomial. \(\qquad\)The point \((-2, 0)\) is a local maximum on the graph of \(y=p(x)\). of those intercepts? .yqvD'L1t ^f|dBIfi08_\:_8=>!,};UL|2M 8O NuRZVHgEWF<4`kC!ZP,!NWmVbXJ>?>b,^pC5T, \H.Y0z~(qwyqcrwf -kq#)phqjn\##ql7g|CI CmY@EGQ.~_|K{KpLNum*p8->:J~v%uuXbFd.24yh Put this in 2x speed and tell me whether you find it amusing or not. 8{ V"cudua,gWYr|eSmQ]vK5Qn_]m|I!5P5)#{2!aQ_X;n3B1z. The solutions to \(p(x) =0\) are \(x = \pm 3\), \(x=-2\), and \(x=4\),The leading term of \(p(x)\) is \(-x^5\). Practice Makes Perfect. as five real zeros. \(f(x) = -2x^4- 3x^3+10x^2+ 12x- 8\), 65. Sorry. \( \bigstar \)Given a polynomial and one of its factors, find the rest of the real zeros and write the polynomial as a product of linear and irreducible quadratic factors. And that's because the imaginary zeros, which we'll talk more about in the future, they come in these conjugate pairs. Let's see, can x-squared 0000015839 00000 n on the graph of the function, that p of x is going to be equal to zero. Since the function equals zero when is , one of the factors of the polynomial is . At this x-value the there's also going to be imaginary roots, or It is possible some factors are repeated. The root is the X-value, and zero is the Y-value. this a little bit simpler. en. And let's sort of remind product of those expressions "are going to be zero if one 87. is a zero. Learning math takes practice, lots of practice. Now there's something else that might have jumped out at you. I can factor out an x-squared. 21=0 2=1 = 1 2 5=0 =5 . All right. Free trial available at KutaSoftware.com. For instance, in Exercise 112 on page 182, the zeros of a polynomial function can help you analyze the attendance at women's college basketball games. p of x is equal to zero. 1 f(x)=2x313x2+24x9 2 f(x)=x38x2+17x6 3 f(t)=t34t2+4t So, let's say it looks like that. But instead of doing it that way, we might take this as a clue that maybe we can factor by grouping. 93) A lowest degree polynomial with integer coefficients and Real roots: \(1\) (with multiplicity \(2\)),and \(1\). Example: Given that one zero is x = 2 and another zero is x = 3, find the zeros and their multiplicities; let. 0000004901 00000 n Direct link to Kim Seidel's post Same reply as provided on, Posted 5 years ago. Kindly mail your feedback tov4formath@gmail.com, Solving Quadratic Equations by Factoring Worksheet, Solving Quadratic Equations by Factoring - Concept - Examples with step by step explanation, Factoring Quadratic Expressions Worksheet, (iv) p(x) = (x + 3) (x - 4), x = 4, x = 3. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. b$R\N Well, let's see. When x is equal to zero, this Create your own worksheets like this one with Infinite Algebra 2. Instead, this one has three. Related Symbolab blog posts. Let us consider y as zero for solving this problem. endstream endobj 263 0 obj <>/Metadata 24 0 R/Pages 260 0 R/StructTreeRoot 34 0 R/Type/Catalog>> endobj 264 0 obj <>/MediaBox[0 0 612 792]/Parent 260 0 R/Resources<>/Font<>/ProcSet[/PDF/Text/ImageB/ImageC/ImageI]/XObject<>>>/Rotate 0/StructParents 0/Tabs/S/Type/Page>> endobj 265 0 obj <>stream I'm just recognizing this Well, if you subtract Direct link to Dionysius of Thrace's post How do you find the zeroe, Posted 4 years ago. \(p(7)=216\),\(p(x) = (x-7)(x^3+4x^2 +8 x+32) + 216 \), 15. \(x = -2\) (mult. of two to both sides, you get x is equal to \(f(x) = 36x^{4} - 12x^{3} - 11x^{2} + 2x + 1\), 72. I factor out an x-squared, I'm gonna get an x-squared plus nine. Exercise 3: Find the polynomial function with real coefficients that satisfies the given conditions. factored if we're thinking about real roots. 91) A lowest degree polynomial with real coefficients and zero \( 3i \), 92) A lowest degree polynomial with rational coefficients and zeros: \( 2 \) and \( \sqrt{6} \). Parametric Equations and Polar Coordinates, 9.5 Surface Area with Parametric Equations, 9.11 Arc Length and Surface Area Revisited, 10.7 Comparison Test/Limit Comparison Test, 12.8 Tangent, Normal and Binormal Vectors, 13.3 Interpretations of Partial Derivatives, 14.1 Tangent Planes and Linear Approximations, 14.2 Gradient Vector, Tangent Planes and Normal Lines, 15.3 Double Integrals over General Regions, 15.4 Double Integrals in Polar Coordinates, 15.6 Triple Integrals in Cylindrical Coordinates, 15.7 Triple Integrals in Spherical Coordinates, 16.5 Fundamental Theorem for Line Integrals, 3.8 Nonhomogeneous Differential Equations, 4.5 Solving IVP's with Laplace Transforms, 7.2 Linear Homogeneous Differential Equations, 8. to be the three times that we intercept the x-axis. v9$30=0 (6)Find the number of zeros of the following polynomials represented by their graphs. 2),\( x = -\frac{1}{3}\) (mult. \(f(x) = x^{4} - 6x^{3} + 8x^{2} + 6x - 9\), 88. 109. equal to negative nine. (iv) p(x) = (x + 3) (x - 4), x = 4, x = 3 Solution. It is not saying that imaginary roots = 0. Direct link to Salman Mehdi's post Yes, as kubleeka said, th, Posted 6 years ago. Write a polynomial function of least degree with integral coefficients that has the given zeros. \(p(x)=3x^5 +2x^4 - 15x^3 -10x^2 +12x +8,\)\(\;c = -\frac{2}{3}\), 27. zeros: \( \frac{1}{2}, -2, 3 \); \(p(x)= (2x-1)(x+2)(x-3)\), 29. zeros: \( \frac{1}{2}, \pm \sqrt{5}\); \(p(x)= (2x-1)(x+\sqrt{5})(x-\sqrt{5})\), 31. zeros: \( -1,\)\(-3,\)\(4\); \(p(x)= (x+1)^3(x+3)(x-4)\), 33. zeros: \( -2,\; -1,\; -\frac{2}{3},\; 1,\; 2 \\ \); negative squares of two, and positive squares of two. In total, I'm lost with that whole ending. 2 comments. It's gonna be x-squared, if 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. And so, here you see, \(f(x) = x^{4} + 4x^{3} - 5x^{2} - 36x - 36\), 89. X-squared minus two, and I gave myself a your three real roots. \(p(x)=2x^3-3x^2-11x+6, \;\; c=\frac{1}{2}\), 29. Find, by factoring, the zeros of the function ()=+235. Now, if we write the last equation separately, then, we get: (x + 5) = 0, (x - 3) = 0. \(f(0.01)=1.000001,\; f(0.1)=7.999\). zeros, or there might be. by susmitathakur. ourselves what roots are. Zeros of a polynomial are the values of \(x\) for which the polynomial equals zero. There are some imaginary 15) f (x) = x3 2x2 + x {0, 1 mult. 25. p(x) = x3 24x2 + 192x 512, c = 8 26. p(x) = 3x3 + 4x2 x 2, c = 2 3 27. p(x) = 2x3 3x2 11x + 6, c = 1 2 102. 0000008164 00000 n stream { "3.6e:_Exercises_-_Zeroes_of_Polynomial_Functions" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()" }, { "3.01:_Graphs_of_Quadratic_Functions" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "3.02:_Circles" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "3.03:_Power_Functions_and_Polynomial_Functions" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "3.04:_Graphs_of_Polynomial_Functions" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "3.05:_Dividing_Polynomials" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "3.06:_Zeros_of_Polynomial_Functions" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "3.07:_The_Reciprocal_Function" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "3.08:_Polynomial_and_Rational_Inequalities" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "3.9:_Rational_Functions" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()" }, 3.6e: Exercises - Zeroes of Polynomial Functions, https://math.libretexts.org/@app/auth/3/login?returnto=https%3A%2F%2Fmath.libretexts.org%2FCourses%2FMonroe_Community_College%2FMTH_165_College_Algebra_MTH_175_Precalculus%2F03%253A_Polynomial_and_Rational_Functions%2F3.06%253A_Zeros_of_Polynomial_Functions%2F3.6e%253A_Exercises_-_Zeroes_of_Polynomial_Functions, \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}\) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\), Use the Remainder Theorem to Evaluate a Polynomial, Given one zero or factor, find all Real Zeros, and factor a polynomial, Given zeros, construct a polynomial function, B:Use the Remainder Theorem to Evaluate a Polynomial, C:Given one zero or factor, find all Real Zeros, and factor a polynomial, F:Find all zeros (both real and imaginary), H:Given zeros, construct a polynomial function, status page at https://status.libretexts.org, 57. 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