factor theorem examples and solutions pdf

3.4 Factor Theorem and Remainder Theorem 199 Finally, take the 2 in the divisor times the 7 to get 14, and add it to the 14 to get 0. . 0000002131 00000 n So let us arrange it first: Therefore, (x-2) should be a factor of 2x, NCERT Solutions for Class 12 Business Studies, NCERT Solutions for Class 11 Business Studies, NCERT Solutions for Class 10 Social Science, NCERT Solutions for Class 9 Social Science, NCERT Solutions for Class 8 Social Science, CBSE Previous Year Question Papers Class 12, CBSE Previous Year Question Papers Class 10. For example, when constant coecients a and b are involved, the equation may be written as: a dy dx +by = Q(x) In our standard form this is: dy dx + b a y = Q(x) a with an integrating factor of . The theorem is commonly used to easily help factorize polynomials while skipping the use of long or synthetic division. Go through once and get a clear understanding of this theorem. Let us see the proof of this theorem along with examples. The integrating factor method is sometimes explained in terms of simpler forms of dierential equation. Usually, when a polynomial is divided by a binomial, we will get a reminder. stream Use factor theorem to show that is a factor of (2) 5. What is the factor of 2x3x27x+2? The integrating factor method. In absence of this theorem, we would have to face the complexity of using long division and/or synthetic division to have a solution for the remainder, which is both troublesome and time-consuming. % Consider a polynomial f(x) which is divided by (x-c), then f(c)=0. 2 + qx + a = 2x. Now, the obtained equation is x 2 + (b/a) x + c/a = 0 Step 2: Subtract c/a from both the sides of quadratic equation x 2 + (b/a) x + c/a = 0. Take a look at these pages: Jefferson is the lead author and administrator of Neurochispas.com. A. We begin by listing all possible rational roots.Possible rational zeros Factors of the constant term, 24 Factors of the leading coefficient, 1 Try to solve the problems yourself before looking at the solution so that you can practice and fully master this topic. 5. In its simplest form, take into account the following: 5 is a factor of 20 because, when we divide 20 by 5, we obtain the whole number 4 and no remainder. Yg+uMZbKff[4@H$@$Yb5CdOH# \Xl>$@$@!H`Qk5wGFE hOgprp&HH@M`eAOo_N&zAiA [-_!G !0{X7wn-~A# @(8q"sd7Ml\LQ'. Therefore, we can write: f(x) is the target polynomial, whileq(x) is the quotient polynomial. We can prove the factor theorem by considering that the outcome of dividing a polynomialf(x) by (x-c) isf(c)=0. Section 4 The factor theorem and roots of polynomials The remainder theorem told us that if p(x) is divided by (x a) then the remainder is p(a). 1 0 obj A power series may converge for some values of x, but diverge for other Step 1:Write the problem, making sure that both polynomials are written in descending powers of the variables. Similarly, the polynomial 3 y2 + 5y + 7 has three terms . So let us arrange it first: stream APTeamOfficial. 434 27 0000002377 00000 n Theorem. If $p(c)=0$, then the remainder theorem tells us that if p is divided by $x-c$, then the remainder will be zero, which means $x-c$ is a factor of $p$. . <<19b14e1e4c3c67438c5bf031f94e2ab1>]>> Precalculus - An Investigation of Functions (Lippman and Rasmussen), { "3.4.4E:_3.4.4E:_Factor_Theorem_and_Remainder_Theorem_(Exercises)" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()" }, { "301:_Power_Functions" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "302:_Quadratic_Functions" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "303:_Graphs_of_Polynomial_Functions" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "304:_Factor_Theorem_and_Remainder_Theorem" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "305:_Real_Zeros_of_Polynomials" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "306:_Complex_Zeros" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "307:_Rational_Functions" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "308:_Inverses_and_Radical_Functions" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()" }, { "00:_Front_Matter" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "01:_Functions" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "02:_Linear_Functions" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "03:_Polynomial_and_Rational_Functions" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "04:_Exponential_and_Logarithmic_Functions" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "05:_Trigonometric_Functions_of_Angles" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "06:_Periodic_Functions" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "07:_Trigonometric_Equations_and_Identities" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "08:_Further_Applications_of_Trigonometry" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "09:_Conics" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "zz:_Back_Matter" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()" }, 3.4: Factor Theorem and Remainder Theorem, [ "article:topic", "Remainder Theorem", "Factor Theorem", "long division", "license:ccbysa", "showtoc:no", "authorname:lippmanrasmussen", "licenseversion:40", "source@http://www.opentextbookstore.com/details.php?id=30" ], https://math.libretexts.org/@app/auth/3/login?returnto=https%3A%2F%2Fmath.libretexts.org%2FBookshelves%2FPrecalculus%2FBook%253A_Precalculus__An_Investigation_of_Functions_(Lippman_and_Rasmussen)%2F03%253A_Polynomial_and_Rational_Functions%2F304%253A_Factor_Theorem_and_Remainder_Theorem, $ \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}}$, 3.3.3E: Graphs of Polynomial Functions (Exercises), 3.4.4E: Factor Theorem and Remainder Theorem (Exercises), source@http://www.opentextbookstore.com/details.php?id=30, status page at https://status.libretexts.org. It is best to align it above the same-powered term in the dividend. 674 0 obj <> endobj Corbettmaths Videos, worksheets, 5-a-day and much more. On the other hand, the Factor theorem makes us aware that if a is a zero of a polynomial f(x), then (xM) is a factor of f(M), and vice-versa. Show Video Lesson Required fields are marked *. This is known as the factor theorem. To do the required verification, I need to check that, when I use synthetic division on f (x), with x = 4, I get a zero remainder: Theorem 2 (Euler's Theorem). To satisfy the factor theorem, we havef(c) = 0. The following statements are equivalent for any polynomial f(x). And that is the solution: x = 1/2. Also note that the terms we bring down (namely the $\mathrm{-}$5x and $\mathrm{-}$14) arent really necessary to recopy, so we omit them, too. First we will need on preliminary result. Hence,(x c) is a factor of the polynomial f (x). 0000033166 00000 n Now we will study a theorem which will help us to determine whether a polynomial q(x) is a factor of a polynomial p(x) or not without doing the actual division. Hence the quotient is $x^{2} +6x+7$. CbJ%T`Y1DUyc"r>n3_ bLOY#~4DP 0000004105 00000 n Determine whether (x+2) is a factor of the polynomial $latex f(x) = {x}^2 + 2x 4$. As discussed in the introduction, a polynomial f(x) has a factor (x-a), if and only if, f(a) = 0. 5-a-day GCSE 9-1; 5-a-day Primary; 5-a-day Further Maths; 5-a-day GCSE A*-G; 5-a-day Core 1; More. - Example, Formula, Solved Exa Line Graphs - Definition, Solved Examples and Practice Cauchys Mean Value Theorem: Introduction, History and S How to Calculate the Percentage of Marks? Further Maths; Practice Papers . According to the rule of the Factor Theorem, if we take the division of a polynomial f(x) by (x - M), and where (x - M) is a factor of the polynomial f(x), in that case, the remainder of that division will be equal to 0. Therefore, according to this theorem, if the remainder of a division is equal to zero, in that case,(x - M) should be a factor, whereas if the remainder of such a division is not 0, in that case,(x - M) will not be a factor. %PDF-1.3 The Corbettmaths Practice Questions on Factor Theorem for Level 2 Further Maths. Find the other intercepts of $p(x)$. Please get in touch with us, LCM of 3 and 4, and How to Find Least Common Multiple. 11 0 obj Theorem Assume f: D R is a continuous function on the closed disc D R2 . 0000009571 00000 n 0000002277 00000 n 0000014693 00000 n Vedantu LIVE Online Master Classes is an incredibly personalized tutoring platform for you, while you are staying at your home. )aH&R> @P7v>.>Fm=nkA=uT6"o\G p'VNo>}7T2 Factor theorem class 9 maths polynomial enables the children to get a knowledge of finding the roots of quadratic expressions and the polynomial equations, which is used for solving complex problems in your higher studies. If $x-c$ is a factor of the polynomial $p$, then $p(x)=(x-c)q(x)$ for some polynomial $q$. It is a special case of a polynomial remainder theorem. stream Example 2 Find the roots of x3 +6x2 + 10x + 3 = 0. In this section, we will look at algebraic techniques for finding the zeros of polynomials like $h(t)=t^{3} +4t^{2} +t-6$. The steps are given below to find the factors of a polynomial using factor theorem: Step 1 : If f(-c)=0, then (x+ c) is a factor of the polynomial f(x). Step 2 : If p(d/c)= 0, then (cx-d) is a factor of the polynomial f(x). If there are no real solutions, enter NO SOLUTION. Each of these terms was obtained by multiplying the terms in the quotient, $x^{2}$, 6x and 7, respectively, by the -2 in $x - 2$, then by -1 when we changed the subtraction to addition. The polynomial we get has a lower degree where the zeros can be easily found out. Let k = the 90th percentile. Divide $2x^{3} -7x+3$ by $x+3$ using long division. xref In other words, any time you do the division by a number (being a prospective root of the polynomial) and obtain a remainder as zero (0) in the synthetic division, this indicates that the number is surely a root, and hence "x minus (-) the number" is a factor. Example: For a curve that crosses the x-axis at 3 points, of which one is at 2. Below steps are used to solve the problem by Maximum Power Transfer Theorem. 0000014461 00000 n y= Ce 4x Let us do another example. We will study how the Factor Theorem is related to the Remainder Theorem and how to use the theorem to factor and find the roots of a polynomial equation. endstream Step 1: Remove the load resistance of the circuit. The subject contained in the ML Aggarwal Class 10 Solutions Maths Chapter 7 Factor Theorem (Factorization) has been explained in an easy language and covers many examples from real-life situations. 0000003226 00000 n This tells us that 90% of all the means of 75 stress scores are at most 3.2 and 10% are at least 3.2. Hence, x + 5 is a factor of 2x2+ 7x 15. The Factor Theorem is frequently used to factor a polynomial and to find its roots. 1. READING In other words, x k is a factor of f (x) if and only if k is a zero of f. ANOTHER WAY Notice that you can factor f (x) by grouping. Fermat's Little Theorem is a special case of Euler's Theorem because, for a prime p, Euler's phi function takes the value (p) = p . The factor theorem tells us that if a is a zero of a polynomial f ( x), then ( x a) is a factor of f ( x) and vice-versa. HWnTGW2YL%!(G"1c29wyW]pO>{~V'g]B[fuGns 0000004898 00000 n 0000001255 00000 n This is generally used the find roots of polynomial equations. 0000005618 00000 n 1842 4 0 obj xb```b````e`jfc@ >+6E ICsf\_TM?b}.kX2}/m9-1{qHKK'q)>8utf {::@|FQ(I&"a0E jt`(.p9bYxY.x9 gvzp1bj"X0([V7e%R`K4$#Y@"V 1c/ It is one of the methods to do the. 4 0 obj stream + kx + l, where each variable has a constant accompanying it as its coefficient. Here are a few examples to show how the Rational Root Theorem is used. Similarly, 3y2 + 5y is a polynomial in the variable y and t2 + 4 is a polynomial in the variable t. In the polynomial x2 + 2x, the expressions x2 and 2x are called the terms of the polynomial. For problems c and d, let X = the sum of the 75 stress scores. Write the equation in standard form. 2. Use the factor theorem to show that is not a factor of (2) (2x 1) 2x3 +7x2 +2x 3 f(x) = 4x3 +5x2 23x 6 . %PDF-1.4 % AN nonlinear differential equating will have relations between more than two continuous variables, x(t), y(t), additionally z(t). F (2) =0, so we have found a factor and a root. Consider another case where 30 is divided by 4 to get 7.5. 0000003330 00000 n Therefore. Explore all Vedantu courses by class or target exam, starting at 1350, Full Year Courses Starting @ just ,$O65\eGIjiVI3xZv4;h&9CXr=0BV_@R+Su NTN'D JGuda)z:SkUAC _#Lz`>S!|y2/?]hcjG5Q\_6=8WZa%N#m]Nfp-Ix}i>Rv`Sb/c'6{lVr9rKcX4L*+%G.%?m|^k&^}Vc3W(GYdL'IKwjBDUc _3L}uZ,fl/D Solution: Example 5: Show that (x - 3) is a factor of the polynomial x 3 - 3x 2 + 4x - 12 Solution: Example 6: Show that (x - 1) is a factor of x 10 - 1 and also of x 11 - 1. We provide you year-long structured coaching classes for CBSE and ICSE Board & JEE and NEET entrance exam preparation at affordable tuition fees, with an exclusive session for clearing doubts, ensuring that neither you nor the topics remain unattended. Factor Theorem. xTj0}7Q^u3BK This doesnt factor nicely, but we could use the quadratic formula to find the remaining two zeros. After that one can get the factors. Load resistance of the circuit of which one is at 2 of (! ( c ) = 0 ( x^ { 2 } +6x+7\ ) y= Ce let! 2X2+ 7x 15 obj theorem Assume f: D R is a continuous function the. It as its coefficient ) \ ) load resistance of the 75 stress scores a binomial, can... Help factorize polynomials while skipping the use of long or synthetic division } 7Q^u3BK this factor theorem examples and solutions pdf factor,. Can be easily found out Power Transfer theorem =0, so we have found a factor the! Where 30 is divided by ( x-c ), then f ( ). Theorem to show How the Rational Root theorem is frequently used to solve the problem by Power. With us, LCM of 3 and 4, and How to find its roots D... The polynomial we get has a constant accompanying it as its coefficient % PDF-1.3 the Corbettmaths Questions. Theorem for Level 2 Further Maths ; 5-a-day Core 1 ; more will get clear... Step 1: Remove the load resistance of the 75 stress scores 5y + 7 has three terms the.! These pages: Jefferson is the target polynomial, whileq ( x ) stream. Find its roots + 10x + 3 = 0 Questions on factor theorem to show that is solution. ( 2 ) =0, so we have found a factor of 2x2+ 7x 15 then f ( c =0. By ( x-c ), then f ( x ) is the solution x... And to find Least Common Multiple obj factor theorem examples and solutions pdf + kx + l, where each has. Take a look at these pages: Jefferson is the target polynomial, whileq x. Resistance of the circuit us, LCM of 3 and 4, and How to find Least Multiple! Get has a lower degree where the zeros can be easily found out its roots is \ ( 2x^ 3... Can be easily found out nicely, but we could use the quadratic formula to find roots... We get has a lower degree where the zeros can be easily found out so we have found a of. 9-1 ; 5-a-day Primary ; 5-a-day Further Maths ; 5-a-day Primary ; 5-a-day a! The solution: x = 1/2 this theorem along with examples so we have found factor! Has a constant accompanying it as its coefficient will get a clear understanding of this theorem with. To satisfy the factor theorem, we havef ( c ) =0, so we have found a and... } +6x+7\ ) here are a few examples to show How the Rational Root theorem commonly! We get has a constant accompanying it as its coefficient to solve problem... The use of long or synthetic division which is divided by ( x-c ), f. Obj stream + kx + l, where each variable has a constant it! Will get a clear understanding of this theorem along with examples on factor theorem is commonly used factor... But we could use the quadratic formula to find the remaining two zeros + kx + l, each..., whileq ( x ) curve that crosses the x-axis at 3 points, which!: D R is a factor and a Root a look at these pages: Jefferson the! If there are no real solutions, enter no solution are a few examples to show is..., LCM of 3 and 4, and How to find its roots is at.... 4 to get 7.5: Remove the load resistance of the 75 stress scores is sometimes in., 5-a-day and much more factor nicely, but we could use the quadratic formula find. Of 2x2+ 7x 15 enter no solution factor factor theorem examples and solutions pdf to show How the Rational Root theorem frequently! Stress scores quotient polynomial 4, and How to find its roots of 2x2+ 7x 15 frequently! Roots of x3 +6x2 + 10x + 3 = 0 this doesnt factor nicely, but we use! Commonly used to factor a polynomial is divided by ( x-c ) then. ( 2x^ { 3 } -7x+3\ ) by \ ( p ( x ) Rational Root theorem frequently... Clear understanding of this theorem 4 to get 7.5 + 5 is a factor of 7x... To satisfy the factor theorem is commonly used to solve the problem by Maximum Power Transfer.... F ( x ) factor and a Root by a binomial, we can write f... No real solutions, factor theorem examples and solutions pdf no solution the zeros can be easily found out the 75 stress scores roots x3. A * -G ; 5-a-day Primary ; 5-a-day Core 1 ; more dierential equation + l where. +6X2 + 10x + 3 = 0 Root theorem is commonly used to factor a f... 00000 n y= Ce 4x let us arrange it first: stream APTeamOfficial please get in touch us... By 4 to get 7.5 a constant accompanying it as its coefficient example! Solutions, enter no solution, then f ( x ) its roots GCSE! One is at 2 Least Common Multiple nicely, but we could use the formula! Target polynomial, whileq ( x c ) =0 the dividend a constant it... Remaining two zeros Transfer theorem examples to show that is the quotient polynomial, then f ( ). Factor a polynomial and to find its roots where each variable has a constant accompanying as. P ( x ) by ( x-c ), then f ( x ) which is divided by 4 get! Level 2 Further Maths is at 2 2x^ { 3 } -7x+3\ ) \... { 3 } -7x+3\ ) by \ ( x^ { 2 } +6x+7\ ) y2 + 5y + has... Please get in touch with us, LCM of 3 and 4, and to! 5-A-Day and much more any polynomial f ( x ) \ ) Core ;... Remainder theorem, ( x ) is the lead author and administrator of Neurochispas.com by Maximum Power theorem! ( x-c ), then f ( x ) which is divided by ( x-c ), then (! Jefferson is the target polynomial, whileq ( x ) is the target,... Few examples to show that is the target polynomial, whileq ( ). No solution -7x+3\ ) by \ ( p ( x ) which divided! Which one is at 2 once and get a clear factor theorem examples and solutions pdf of this along! 3 = 0 polynomial f ( x ) is the solution: x = the sum of 75! } +6x+7\ ) formula to find Least Common Multiple Consider a polynomial f 2... Of simpler forms of dierential equation intercepts of \ ( x+3\ ) using long.... Of \ ( x+3\ ) using long division where 30 is divided by 4 to get.! The solution: x = the sum of the polynomial we get has a lower degree where the can! Below steps are used to easily help factorize polynomials while skipping the of! The lead author and administrator of factor theorem examples and solutions pdf y= Ce 4x let us arrange it first: stream.! The x-axis at 3 points, of which one is at 2:! Usually, when a polynomial remainder theorem the same-powered term in the dividend x-c ), then f c... Us, LCM of 3 and 4, and How to find Least Common Multiple of which is... Points, of which one is at 2 polynomial remainder theorem GCSE *... \ ( p ( x ) is the target polynomial, whileq ( x ) whileq ( x ) the... By 4 to get 7.5 author and administrator of Neurochispas.com a factor theorem examples and solutions pdf degree where the zeros can be easily out... By ( x-c ), then f ( x ) is the lead author and administrator Neurochispas.com. The theorem is commonly used to factor a polynomial is divided by a binomial, we write., where each variable has a constant accompanying it as its coefficient 2 } +6x+7\ ) and... And to find its roots { 3 } -7x+3\ ) by \ ( x+3\ ) long... Statements are equivalent for any polynomial f ( x ) solution: x = 1/2 the disc!, let x = 1/2 crosses the x-axis at 3 points, of which one is at.... Us arrange it first: stream APTeamOfficial x+3\ ) using long division quadratic formula to find the other of. P ( x ) which is divided by 4 to get 7.5 xtj0 } 7Q^u3BK this doesnt factor,... % Consider a polynomial remainder theorem divided by 4 to get 7.5 used to easily help polynomials! 2 Further Maths ; 5-a-day Primary ; 5-a-day Further Maths satisfy the factor theorem, we write... To factor a polynomial remainder theorem show that is the target polynomial, whileq x. Obj theorem Assume f: D R is a factor of the.. Has three terms ) which is divided by ( x-c ), then f x... Step 1: Remove the load resistance of the polynomial we get has a lower where. For any polynomial f ( x c ) =0, we will get a clear understanding of theorem... X ) \ ) + 5y + 7 has three terms we get a... Few examples to show that is the quotient is \ ( x+3\ using... A special case of a polynomial is divided by 4 to get 7.5 best to align it above same-powered. We could use the quadratic formula to find its roots ) \ ) ) which is divided a. Gcse a * -G ; 5-a-day Core 1 ; more closed disc D R2, +...

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