factor theorem examples and solutions pdf

6. xw`g. We have grown leaps and bounds to be the best Online Tuition Website in India with immensely talented Vedantu Master Teachers, from the most reputed institutions. Algebraic version. Rather than finding the factors by using polynomial long division method, the best way to find the factors are factor theorem and synthetic division method. As result,h(-3)=0 is the only one satisfying the factor theorem. %%EOF Geometric version. 0000002952 00000 n <>/Font<>/ProcSet[/PDF/Text/ImageB/ImageC/ImageI] >>/MediaBox[ 0 0 612 792] /Contents 4 0 R/Group<>/Tabs/S/StructParents 0>> 0000007401 00000 n 6x7 +3x4 9x3 6 x 7 + 3 x 4 9 x 3 Solution. Hence, x + 5 is a factor of 2x2+ 7x 15. 6 0 obj \3;e". Examples Example 4 Using the factor theorem, which of the following are factors of 213 Solution Let P(x) = 3x2 2x + 3 3x2 Therefore, Therefore, c. PG) . This theorem is mainly used to easily help factorize polynomials without taking the help of the long or the synthetic division process. There is another way to define the factor theorem. Hence the possibilities for rational roots are 1, 1, 2, 2, 4, 4, 1 2, 1 2, 1 3, 1 3, 2 3, 2 3, 4 3, 4 3. %HPKm/"OcIwZVjg/o&f]gS},L&Ck@}w> G35v&0` Y_uf>X%nr)]4epb-!>;,I9|3gIM_bKZGGG(b [D&F e`485X," s/ ;3(;a*g)BdC,-Dn-0vx6b4 pdZ eS` ?4;~D@ U x2(26x)+4x(412x) x 2 ( 2 6 x . 0000006640 00000 n We have constructed a synthetic division tableau for this polynomial division problem. stream Attempt to factor as usual (This is quite tricky for expressions like yours with huge numbers, but it is easier than keeping the a coeffcient in.) Please get in touch with us, LCM of 3 and 4, and How to Find Least Common Multiple. Then f is constrained and has minimal and maximum values on D. In other terms, there are points xm, aM D such that f (x_ {m})\leq f (x)\leq f (x_ {M}) \)for each feasible point of x\inD -----equation no.01. In case you divide a polynomial f(x) by (x - M), the remainder of that division is equal to f(c). Rs 9000, Learn one-to-one with a teacher for a personalised experience, Confidence-building & personalised learning courses for Class LKG-8 students, Get class-wise, author-wise, & board-wise free study material for exam preparation, Get class-wise, subject-wise, & location-wise online tuition for exam preparation, Know about our results, initiatives, resources, events, and much more, Creating a safe learning environment for every child, Helps in learning for Children affected by 0000027213 00000 n Lets see a few examples below to learn how to use the Factor Theorem. 0 To learn the connection between the factor theorem and the remainder theorem. Rewrite the left hand side of the . 2 - 3x + 5 . Divide by the integrating factor to get the solution. Likewise, 3 is not a factor of 20 because, when we are 20 divided by 3, we have 6.67, which is not a whole number. Then Bring down the next term. Determine whether (x+2) is a factor of the polynomial $latex f(x) = {x}^2 + 2x 4$. e 2x(y 2y)= xe 2x 4. 0000015909 00000 n 2 + qx + a = 2x. First, equate the divisor to zero. What is Simple Interest? xb```b``;X,s6 y Keep visiting BYJUS for more information on polynomials and try to solve factor theorem questions from worksheets and also watch the videos to clarify the doubts. In terms of algebra, the remainder factor theorem is in reality two theorems that link the roots of a polynomial following its linear factors. Factor theorem is frequently linked with the remainder theorem. So linear and quadratic equations are used to solve the polynomial equation. <> Step 1: Check for common factors. We begin by listing all possible rational roots.Possible rational zeros Factors of the constant term, 24 Factors of the leading coefficient, 1 0000027699 00000 n 0000036243 00000 n First we will need on preliminary result. Show Video Lesson Now, multiply that \(x^{2}\) by \(x-2\) and write the result below the dividend. 5. has the integrating factor IF=e R P(x)dx. As discussed in the introduction, a polynomial f (x) has a factor (x-a), if and only if, f (a) = 0. Moreover, an evaluation of the theories behind the remainder theorem, in addition to the visual proof of the theorem, is also quite useful. Doing so gives, Since the dividend was a third degree polynomial, the quotient is a quadratic polynomial with coefficients 5, 13 and 39. with super achievers, Know more about our passion to This gives us a way to find the intercepts of this polynomial. First, we have to test whether (x+2) is a factor or not: We can start by writing in the following way: now, we can test whetherf(c) = 0 according to the factor theorem: Given thatf(-2) is not equal to zero, (x+2) is not a factor of the polynomial given. Next, take the 2 from the divisor and multiply by the 1 that was "brought down" to get 2. Note that is often instead required to be open but even under such an assumption, the proof only uses a closed rectangle within . Interested in learning more about the factor theorem? Step 1:Write the problem, making sure that both polynomials are written in descending powers of the variables. 2. The techniques used for solving the polynomial equation of degree 3 or higher are not as straightforward. Factor theorem is frequently linked with the remainder theorem, therefore do not confuse both. This theorem is used primarily to remove the known zeros from polynomials leaving all unknown zeros unimpaired, thus by finding the zeros easily to produce the lower degree polynomial. So let us arrange it first: pdf, 43.86 MB. 0000007800 00000 n Rational Root Theorem Examples. I used this with my GCSE AQA Further Maths class. Factor four-term polynomials by grouping. 0000006280 00000 n 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. x nH@ w Resource on the Factor Theorem with worksheet and ppt. Through solutions, we can nd ideas or tech-niques to solve other problems or maybe create new ones. For problems c and d, let X = the sum of the 75 stress scores. xbbRe`b``3 1 M 434 27 the factor theorem If p(x) is a nonzero polynomial, then the real number c is a zero of p(x) if and only if x c is a factor of p(x). 0000030369 00000 n The factor theorem can be used as a polynomial factoring technique. 0000000851 00000 n 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. The reality is the former cant exist without the latter and vice-e-versa. To learn how to use the factor theorem to determine if a binomial is a factor of a given polynomial or not. 0000033438 00000 n In mathematics, factor theorem is used when factoring the polynomials completely. u^N{R YpUF_d="7/v(QibC=S&n\73jQ!f.Ei(hx-b_UG l}e4W[;E#xmX$BQ The first three numbers in the last row of our tableau are the coefficients of the quotient polynomial. 0000018505 00000 n f (1) = 3 (1) 4 + (1) 3 (1)2 +3 (1) + 2, Hence, we conclude that (x + 1) is a factor of f (x). (iii) Solution : 3x 3 +8x 2-6x-5. The possibilities are 3 and 1. r 1 6 10 3 3 1 9 37 114 -3 1 3 1 0 There is a root at x = -3. If f(x) is a polynomial, then x-a is the factor of f(x), if and only if, f(a) = 0, where a is the root. Find the other intercepts of \(p(x)\). % Further Maths; Practice Papers . @8hua hK_U{S~$[fSa&ac|4K)Y=INH6lCKW{p I#K(5@{/ S.|`b/gvKj?PAzm|*UvA=~zUp4-]m`vrmp`8Vt9bb]}9_+a)KkW;{z_+q;Ev]_a0` ,D?_K#GG~,WpJ;z*9PpRU )9K88/<0{^s$c|\Zy)0p x5pJ YAq,_&''M$%NUpqgEny y1@_?8C}zR"$,n|*5ms3wpSaMN/Zg!bHC{p\^8L E7DGfz8}V2Yt{~ f:2 KG"8_o+ %PDF-1.3 In other words. %PDF-1.3 Find out whether x + 1 is a factor of the below-given polynomial. Example 1: Finding Rational Roots. In division, a factor refers to an expression which, when a further expression is divided by this particular factor, the remainder is equal to zero (0). @\)Ta5 << /Length 5 0 R /Filter /FlateDecode >> px. If (x-c) is a factor of f(x), then the remainder must be zero. As discussed in the introduction, a polynomial f(x) has a factor (x-a), if and only if, f(a) = 0. 0000001255 00000 n The horizontal intercepts will be at \((2,0)\), \(\left(-3-\sqrt{2} ,0\right)\), and \(\left(-3+\sqrt{2} ,0\right)\). 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. Your Mobile number and Email id will not be published. The algorithm we use ensures this is always the case, so we can omit them without losing any information. Determine which of the following polynomial functions has the factor(x+ 3): We have to test the following polynomials: Assume thatx+3 is a factor of the polynomials, wherex=-3. 1. startxref From the previous example, we know the function can be factored as \(h(x)=\left(x-2\right)\left(x^{2} +6x+7\right)\). Welcome; Videos and Worksheets; Primary; 5-a-day. What is the factor of 2x3x27x+2? 7.5 is the same as saying 7 and a remainder of 0.5. << /ProcSet [ /PDF /Text /ImageB /ImageC /ImageI ] /ColorSpace << /Cs2 9 0 R Multiplying by -2 then by -1 is the same as multiplying by 2, so we replace the -2 in the divisor by 2. These two theorems are not the same but dependent on each other. Usually, when a polynomial is divided by a binomial, we will get a reminder. A power series may converge for some values of x, but diverge for other 0000002710 00000 n 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. endstream endobj 459 0 obj <>/Size 434/Type/XRef>>stream 0 Particularly, when put in combination with the rational root theorem, this provides for a powerful tool to factor polynomials. Solution: To solve this, we have to use the Remainder Theorem. 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. Theorem Assume f: D R is a continuous function on the closed disc D R2 . 9Z_zQE Sub- #}u}/e>3aq. 0000002236 00000 n 0000003855 00000 n 0000007948 00000 n % stream The integrating factor method. Synthetic division is our tool of choice for dividing polynomials by divisors of the form \(x - c\). We will not prove Euler's Theorem here, because we do not need it. This result is summarized by the Factor Theorem, which is a special case of the Remainder Theorem. Hence,(x c) is a factor of the polynomial f (x). Heaviside's method in words: To determine A in a given partial fraction A s s 0, multiply the relation by (s s 0), which partially clears the fraction. Also take note that when a polynomial (of degree at least 1) is divided by \(x - c\), the result will be a polynomial of exactly one less degree. >> To find the remaining intercepts, we set \(4x^{2} -12=0\) and get \(x=\pm \sqrt{3}\). \[x=\dfrac{-6\pm \sqrt{6^{2} -4(1)(7)} }{2(1)} =-3\pm \sqrt{2} \nonumber \]. Then \(p(c)=(c-c)q(c)=0\), showing \(c\) is a zero of the polynomial. 0000008973 00000 n endobj <> 0000000016 00000 n Since dividing by \(x-c\) is a way to check if a number is a zero of the polynomial, it would be nice to have a faster way to divide by \(x-c\) than having to use long division every time. By factor theorem, if p(-1) = 0, then (x+1) is a factor of p(x) = 2x 4 +9x 3 +2x 2 +10x+15. 0000001945 00000 n This proves the converse of the theorem. The Remainder Theorem Date_____ Period____ Evaluate each function at the given value. It basically tells us that, if (x-c) is a factor of a polynomial, then we must havef(c)=0. Factor Theorem: Polynomials An algebraic expression that consists of variables with exponents as whole numbers, coefficients, and constants combined using basic mathematical operations like addition, subtraction, and multiplication is called a polynomial. The factor theorem states that a polynomial has a factor provided the polynomial x - M is a factor of the polynomial f(x) island provided f f (M) = 0. Substitute the values of x in the equation f(x)= x2+ 2x 15, Since the remainders are zero in the two cases, therefore (x 3) and (x + 5) are factors of the polynomial x2+2x -15. Solving the equation, assume f(x)=0, we get: Because (x+5) and (x-3) are factors of x2 +2x -15, -5 and 3 are the solutions to the equation x2 +2x -15=0, we can also check these as follows: If the remainder is zero, (x-c) is a polynomial of f(x). Required fields are marked *. rnG In its basic form, the Chinese remainder theorem will determine a number p p that, when divided by some given divisors, leaves given remainders. Let us see the proof of this theorem along with examples. Solved Examples 1. Each example has a detailed solution. 0000004440 00000 n Some bits are a bit abstract as I designed them myself. window.__mirage2 = {petok:"_iUEwVe.LVVWL1qoF4bc2XpSFh1TEoslSEscivdbGzk-31536000-0"}; Example 1 Solve for x: x3 + 5x2 - 14x = 0 Solution x(x2 + 5x - 14) = 0 \ x(x + 7)(x - 2) = 0 \ x = 0, x = 2, x = -7 Type 2 - Grouping terms With this type, we must have all four terms of the cubic expression. To find the solution of the function, we can assume that (x-c) is a polynomial factor, wherex=c. The general form of a polynomial is axn+ bxn-1+ cxn-2+ . Factor theorem is a polynomial remainder theorem that links the factors of a polynomial and its zeros together. To use synthetic division, along with the factor theorem to help factor a polynomial. Find the solution of y 2y= x. 0000002874 00000 n endobj Then for each integer a that is relatively prime to m, a(m) 1 (mod m). To divide \(x^{3} +4x^{2} -5x-14\) by \(x-2\), we write 2 in the place of the divisor and the coefficients of \(x^{3} +4x^{2} -5x-14\)in for the dividend. The Corbettmaths Practice Questions on Factor Theorem for Level 2 Further Maths. Factor P(x) = 6x3 + x2 15x + 4 Solution Note that the factors of 4 are 1,-1, 2,-2,4,-4, and the positive factors of 6 are 1,2,3,6. 0000003030 00000 n First, lets change all the subtractions into additions by distributing through the negatives. << /Length 5 0 R /Filter /FlateDecode >> Step 2:Start with 3 4x 4x2 x Step 3:Subtract by changing the signs on 4x3+ 4x2and adding. << /Length 12 0 R /Type /XObject /Subtype /Image /Width 681 /Height 336 /Interpolate Factor Theorem. zZBOeCz&GJmwQ-~N1eT94v4(fL[N(~l@@D5&3|9&@0iLJ2x LRN+.wge%^h(mAB hu.v5#.3}E34;joQTV!a:= Use the factor theorem detailed above to solve the problems. 0000001441 00000 n )aH&R> @P7v>.>Fm=nkA=uT6"o\G p'VNo>}7T2 Using the graph we see that the roots are near 1 3, 1 2, and 4 3. Consider 5 8 4 2 4 16 4 18 8 32 8 36 5 20 5 28 4 4 9 28 36 18 . Factor Theorem Factor Theorem is also the basic theorem of mathematics which is considered the reverse of the remainder theorem. 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. If the term a is any real number, then we can state that; (x a) is a factor of f (x), if f (a) = 0. 2. 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. Problem 5: If two polynomials 2x 3 + ax 2 + 4x - 12 and x 3 + x 2 -2x +a leave the same remainder when divided by (x - 3), find the value of a, and what is the remainder value? The interactive Mathematics and Physics content that I have created has helped many students. -3 C. 3 D. -1 endstream endobj 718 0 obj<>/W[1 1 1]/Type/XRef/Index[33 641]>>stream 0000004362 00000 n 0000005618 00000 n Concerning division, a factor is an expression that, when a further expression is divided by this factor, the remainder is equal to zero (0). Start by writing the problem out in long division form. Factor theorem is commonly used for factoring a polynomial and finding the roots of the polynomial. 0000003330 00000 n Since, the remainder = 0, then 2x + 1 is a factor of 4x3+ 4x2 x 1, Check whetherx+ 1 is a factor of x6+ 2x (x 1) 4, Now substitute x = -1 in the polynomial equation x6+ 2x (x 1) 4 (1)6 + 2(1) (2) 4 = 1Therefore,x+ 1 is not a factor of x6+ 2x (x 1) 4. 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). Therefore,h(x) is a polynomial function that has the factor (x+3). Use Algebra to solve: A "root" is when y is zero: 2x+1 = 0. >zjs(f6hP}U^=`W[wy~qwyzYx^Pcq~][+n];ER/p3 i|7Cr*WOE|%Z{\B| For example - we will get a new way to compute are favorite probability P(~as 1st j~on 2nd) because we know P(~on 2nd j~on 1st). After that one can get the factors. That being said, lets see what the Remainder Theorem is. The method works for denominators with simple roots, that is, no repeated roots are allowed. 0000014461 00000 n Lets take a moment to remind ourselves where the \(2x^{2}\), \(12x\) and 14 came from in the second row. endobj Hence, the Factor Theorem is a special case of Remainder Theorem, which states that a polynomial f (x) has a factor x a, if and only if, a is a root i.e., f (a) = 0. 5 0 obj \[x^{3} +8=(x+2)\left(x^{2} -2x+4\right)\nonumber \]. Emphasis has been set on basic terms, facts, principles, chapters and on their applications. We use 3 on the left in the synthetic division method along with the coefficients 1,2 and -15 from the given polynomial equation. The theorem is commonly used to easily help factorize polynomials while skipping the use of long or synthetic division. Determine whetherx+ 1 is a factor of the polynomial 3x4+x3x2+ 3x+ 2, Substitute x = -1 in the equation; 3x4+x3x2+ 3x+ 2. 3(1)4 + (1)3 (1)2 +3(1) + 2= 3(1) + (1) 1 3 + 2 = 0Therefore,x+ 1 is a factor of 3x4+x3x2+ 3x+ 2, Check whether 2x + 1 is a factor of the polynomial 4x3+ 4x2 x 1. Theorem 2 (Euler's Theorem). trailer endobj endobj If x + 4 is a factor, then (setting this factor equal to zero and solving) x = 4 is a root. m 5gKA6LEo@`Y&DRuAs7dd,pm3P5)$f1s|I~k>*7!z>enP&Y6dTPxx3827!'\-pNO_J. Application Of The Factor Theorem How to peck the factor theorem to ache if x c is a factor of the polynomial f Examples fx. Let us take the following: 5 is a factor of 20 since, when we divide 20 by 5, we get the whole number 4 and there is no remainder. y= Ce 4x Let us do another example. Similarly, 3 is not a factor of 20 since when we 20 divide by 3, we have 6.67, and this is not a whole number. 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.

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