Chapter 483Partial FractionChapter 4Partial Fractions4.1 Introduction: A fraction is a symbol indicating the division ofintegers. For example,13 2, are fractions and are called Common9 3Fraction. The dividend (upper number) is called the numerator N(x) andthe divisor (lower number) is called the denominator, D(x).From the previous study of elementary algebra we have learnt howthe sum of different fractions can be found by taking L.C.M. and then addall the fractions. For examplei)ii )Here we study the reverse process, i.e., we split up a single fraction into anumber of fractions whose denominators are the factors of denominator ofthat fraction. These fractions are called Partial fractions.4.2Partial fractions :To express a single rational fraction into the sum of two or moresingle rational fractions is called Partial fraction resolution.For example,2x x 2 1 111 2x x 1 x 1x(x 1)1112x x 2 1 istheresultantfractionandare itsx x 1 x 1x(x 2 1)partial fractions.4.3Polynomial:Any expression of the form P(x) anxn an-1 xn-1 . a2x2 a1x a0 where an, an-1, ., a2, a1, a0 are real constants, if an 0 then P(x)is called polynomial of degree n.4.4Rational fraction:p, q 0 is called a rational number. SimilarlyqN(x)the quotient of two polynomialswhere D(x) 0 , with no commonD(x)We know thatfactors, is called a rational fraction. A rational fraction is of two types:
Chapter 44.584Partial FractionProper Fraction:A rational fractionN(x)is called a proper fraction if the degreeD(x)of numerator N(x) is less than the degree of Denominator D(x).For example(i)(ii)4.69x 2 9x 6(x 1)(2x 1)(x 2)6x 273x 3 9xImproper Fraction:A rational fractionN(x)is called an improper fraction if theD(x)degree of the Numerator N(x) is greater than or equal to the degree of theDenominator D(x)For example(i)(ii)2x 3 5x 2 3x 10x2 16x 3 5x 2 73x 2 2x 1Note: An improper fraction can be expressed, by division, as the sum of apolynomial and a proper fraction.For example:8x 46x 3 5x 2 7 (2x 3) 22x 2x 13x 2x 1Which is obtained as, divide 6x2 5x2 – 7 by 3x2 – 2x – 1 then weget a polynomial (2x 3) and a proper fraction4.78x 4x 2x 12Process of Finding Partial Fraction:A proper fraction(I)N(x)can be resolved into partial fractions as:D(x)If in the denominator D(x) a linear factor (ax b) occurs and isnon-repeating, its partial fraction will be of the formA,where A is a constant whose value is to be determined.ax b
Chapter 4(II)85Partial FractionIf in the denominator D(x) a linear factor (ax b) occurs ntimes, i.e., (ax b)n, then there will be n partial fractions of theformA3A1A2An . 23ax b (ax b)(ax b)(ax b)n,where A1, A2, A3 - - - - - - - - - An are constants whose valuesare to be determined(III)If in the denominator D(x) a quadratic factor ax2 bx coccurs and is non-repeating, its partial fraction will be of the formAx B, where A and B are constants whose values are toax bx c2be determined.(IV)If in the denominator a quadratic factor ax2 bx coccurs n times, i.e., (ax2 bx c)n ,then there will be n partialfractions of the formA3x B3A1x B1A 2 x B2 222ax bx c (ax bx c)(ax 2 bx c)3A n x Bn- - - - - - - - - - (ax 2 bx c)nWhere A1, A2, A3 - - - - - - - - An and B1, B2, B3 - - - - - - - Bn areconstants whose values are to be determined.Note: The evaluation of the coefficients of the partial fractions is basedon the following theorem:If two polynomials are equal for all values of the variables, then thecoefficients having same degree on both sides are equal, for example , ifpx2 qx a 2x2 – 3x 5 x , thenp 2, q - 3 and a 5.4.8Type IWhen the factors of the denominator are all linear and distinct i.e.,non repeating.Example 1:ResolveSolution:7x 25into partial fractions.(x 3)(x 4)AB7x 25 ------------------(1)(x 3)(x 4) x 3 x 4Multiplying both sides by L.C.M. i.e., (x – 3)(x – 4), we get7x – 25 A(x – 4) B(x – 3) -------------- (2)7x – 25 Ax – 4A Bx – 3B
Chapter 486Partial Fraction7x – 25 Ax Bx – 4A – 3B7x – 25 (A B)x – 4A – 3BComparing the co-efficients of like powers of x on both sides, wehave7 A B and–25 – 4A – 3BSolving these equation we getA 4 andB 3Hence the required partial fractions are:7x 2543 (x 3)(x 4) x 3 x 4Alternative Method:Since 7x – 25 A(x – 4) B(x – 3)Putx -4 0, x 4 in equation (2)7(4) – 25 A(4 – 4) B(4 – 3)28 – 25 0 B(1)B 3Put x – 3 0 x 3 in equation (2)7(3) – 25 A(3 – 4) B(3 – 3)21 – 25 A(–1) 0–4 –AA 4Hence the required partial fractions are7x 2543 (x 3)(x 4) x 3 x 4Note : The R.H.S of equation (1) is the identity equation of L.H.SExample 2:7x 25(x 3)(x 4)7x 25Solution : The identity equation ofis(x 3)(x 4)7x 25 (x 3)(x 4)write the identity equation ofExample 3:1Resolve into partial fraction: x2 - 1
Chapter 4871AB x -1x-1x 1Solutios:1Partial Fraction2 A(x 1) B (x – 1)x – 1 0,Put1 A (1 1) B(1 – 1)Putx 1 0,(1) x 1 in equation (1) A x -1 in equation (1)121 A (-1 1) B (-1-1)1 -2B, 1B 2111x2 - 1 2(x - 1) - 2(x 1)Example 4:Resolve into partial fractions6x3 5x 2 73x 2 2x 1Solution:This is an improper fraction first we convert it into a polynomialand a proper fraction by division.Let6x 3 5x 2 78x 4 (2x 3) 223x 2x 1x 2x 18x 48x 4AB 2x 2x 1 (3x 1) x 1 3x 1Multiplying both sides by (x – 1)(3x 1) we get8x – 4 A(3x 1) B(x – 1)Putx – 1 0, x 1 in (I), we getThe value of A8(1) – 4 A(3(1) 1) B(1 – 1)8 – 4 A(3 1) 04 4AA 1 Put 3x 1 0 x 1in (I)3(I)
Chapter 4 88Partial Fraction 1 1 8 4 B 1 3 3 8 4 4 3 3 204 B3320 3B x 534Hence the required partial fractions are6x 3 5x 2 715 (2x 3) x 1 3x 13x 2 2x 1Example 5:8x 8x 3 2x 2 8x8x 88x 88x 8 322x 2x 8x x(x 2x 8) x(x 4)(x 2)8x 8ABC x 3 2x 2 8x x x 4 x 2Resolve into partial fractionSolution:LetMultiplying both sides by L.C.M. i.e., x(x – 4)(x 2)8x – 8 A(x – 4)(x 2) Bx(x 2) Cx(x – 4)(I)Put x 0 in equation (I), we have8 (0) – 8 A(0 – 4)(0 2) B(0)(0 2) C(0)(0 – 4)–8 –8A 0 0A 1 Put x – 4 0 x 4 in Equation (I), we have8 (4) – 8 B (4) (4 2)32 – 8 24B24 24BB 1 Put x 2 0 x – 2 in Eq. (I), we have8(–2) –8 C(–2)( –2 –4)–16 –8 C(–2)( –6)–24 12CC –2 Hence the required partial fractions
Chapter 489Partial Fraction8x 8112 2x 2x 8x x x 4 x 23Exercise 4.1Resolve into partial fraction:Q.12x 3(x 2)(x 5)Q.22x 5x 5x 6Q.33x 2 2x 5(x 2)(x 2)(x 3)Q.4(x 1)(x 2)(x 3)(x 4)(x 5)(x 6)Q.5x(x a)(x b)(x c)Q.61(1 ax)(1 bx)(1 cx)Q.7Q.9Q.112x 3 x 2 1(x 3)(x 1)(x 5)6x 274x 3 9xx4(x 1)(x 2)(x 3)21(1 x)(1 2x)(1 3x)9x 2 9x 6Q.10(x 1)(2x 1)(x 2)Q.82x 3 x 2 x 3Q.12x(x 1)(2x 3)Answers 4.1Q.111 x 2 x 5Q.211 x 2 x 331128 20(x 2) 4(x 2) 5(x 3)324301 Q.4x 4 x 5 x 6abcQ.5 (a b)(a c)(x a) (b a)(b c)(x b) (c b)(c a)(x c)a2b2c2 Q.6(a b)(a c)(1 ax) (b a)(b c)(1 bx) (c b)(c a)(1 cx)Q.3
Chapter 490Partial Fraction311137 4(x 3) 12(x 1) 6(x 5)149Q.8 2(1 x) (1 2x) 2(1 3x)342 Q.9x 2x 3 2x 3234 Q.10x 1 2x 1 x 1211681Q.11 x 6 2(x 1) x 2 2(x 3)118Q.12 1 x 5(x 1) 5(2x 3)Q.72 4.9Type II:When the factors of the denominator are all linear but some arerepeated.Example 1:x 2 3x 1Resolve into partial fractions:(x 1)2 (x 2)Solution:x 2 3x 1ABC 22x 2(x 1) (x 2) x 1 (x 1)Multiplying both sides by L.C.M. i.e., (x – 1)2 (x – 2), we getx2 – 3x 1 A(x – 1)(x – 2) B(x – 2) C(x – 1)2 (I)Putting x – 1 0x 1 in (I), then 2(1) – 3 (1) 1 B (1 – 2)1 – 3 1 –B–1 –BB 1 x 2 in (I), thenPutting x – 2 0 (2)2 – 3 (2) 1 C (2 – 1)24 – 6 1 C(1)2–1 C 2Now x – 3x 1 A(x2 – 3x 2) B(x – 2) C(x2 – 2x 1)Comparing the co-efficient of like powers of x on both sides, we getA C 1A 1–C
Chapter 491Partial Fraction 1 – (– 1) 1 1 2A 2 Hence the required partial fractions arex 2 3x 1211 22x 2(x 1) (x 2) x 1 (x 1)Example 2:Resolve into partial fraction1x (x 1)4Solution1A B C DE 2 3 4 x 1x (x 1) x xxx4Where A, B, C, D and E are constants. To find these constantsmultiplying both sides by L.C.M. i.e., x4 (x 1), we get1 A(x3)(x 1) Bx2 (x 1) Cx (x 1) D(x 1) Ex4(I)Puttingx 1 in Eq. (I)1 E( 1)4E 1 Putting x 0 in Eq. (I), we have1 D(0 1)1 DD 1 1 A(x4 x3) B(x3 x2) C(x2 x) D(x 1) ExComparing the co-efficient of like powers of x on both sides.Co-efficient of x3 : A B 0 (i)Co-efficient of x2 : B C 0 (ii)Co-efficient of x : C D 0 (iii)Putting the value of D 1 in (iii)C 1 0C 1 Putting this value in (ii), we getB–1 0B 1 Putting B 1 in (i), we haveA 1 0A 1
Chapter 492Partial FractionHence the required partial fraction are1 1 1111 x 4 (x 1) x x 2 x 3 x 4 x 1Example 3:Resolve into partial fractions4 7x(2 3x)(1 x)2Solution:4 7xABC 22 3x 1 x (1 x)2(2 3x)(1 x)Multiplying both sides by L.C.M. i.e., (2 3x) (1 x)2We get4 7x A(1 x)2 B(2 3x)(1 x) C(2 3x) . (I)Put 2 3x 0Then x 2 2 4 7 A 1 3 3 14 1 4 A 3 3 2 1 A3 9 2 9A x 63 12in (I)322A –61 x 0x –1 in eq. (I), we get 4 7 (–1) C ( 2 – 3)4 – 7 C(–1)–3 –CC 3 4 7x A(x2 2x 1) B(2 5x 3x2) C(2 3x)Comparing the co-efficient of x2 on both sidesA 3B 0– 6 3B 03B 6B 2 Hence the required partial fraction will bePut 623 2 3x 1 x (1 x)2
Chapter 493Partial FractionExercise 4.2Resolve into partial fraction:Q.1x 4(x 2)2 (x 1)Q2.1(x 1)(x 2 1)Q.34x 3(x 1)2 (x 2 1)Q.42x 1(x 3)(x 1)(x 2)2Q.56x 2 11x 32(x 6)(x 1)2Q.6x2 x 3(x 1)3Q.84x 2 13x(x 3)(x 2)2Q.10x 3 8x 2 17x 1(x 3)3Q.122x 1(x 2)(x 3)2Q.7Q.9Q.115x 2 36x 27x 4 6x 3 9x 2x4 1x 2 (x 1)x2(x 1)3 (x 2)Answers4.2Q.1Q.2Q.3Q.4Q.5Q.6Q.7Q.8 111 4(x 1) 4(x 1) 2(x 1)21752 22(x 1) 2(x 1) (x 1)(x 1)35141 4(x 3) 12(x 1) 3(x 2) (x 2)21043 x 6 x 1 (x 1)2113 2x 1 (x 1)(x 1)32 3214 2 x x(x 3) (x 3)2312 x 3 x 2 (x 2)2
Chapter 494Partial Fraction1 12 2 x xx 1147Q.10 1 2x 3 (x 3)(x 3)34514Q.11 2327(x 1) 9(x 1)3(x 1) 27(x 2)337Q.12 25(x 2) 25(x 3) 5(x 3)2Q.9x 1 4.10Type III:When the denominator contains ir-reducible quadratic factorswhich are non-repeated.Example 1:Resolve into partial fractionsSolution:9x 7(x 3)(x 2 1)9x 7ABx C (x 3)(x 2 1) x 3 x 2 1Multiplying both sides by L.C.M. i.e., (x 3)(x2 1), we get9x – 7 A(x2 1) (Bx C)(x 3)(I)Putx 3 0x –3 in Eq. (I), we have 9(–3) –7 A((–3)2 1) (B(–3) C)(–3 3)–27 –7 10A 034A 10 A 1759x – 7 A(x2 1) B(x2 3x) C(x 3)Comparing the co-efficient of like powers of x on both sidesA B 03B C 9Putting value of A in Eq. (i) 17 B 05 B 175From Eq. (iii) 17 4 C 9 – 3B 9 – 3 9 515 C 65
Chapter 495Partial FractionHence the required partial fraction are 1717x 6 5(x 3) 5(x 2 1)Example 2:x2 1Resolve into partial fraction 4x x2 1Solution:Letx2 1x2 1 x 4 x 2 1 (x 2 x 1)(x 2 x 1)x2 1Ax BCx D (x 2 x 1)(x 2 x 1) (x 2 x 1) (x 2 x 1)Multiplying both sides by L.C.M. i.e., (x 2 x 1)(x 2 x 1)x2 1 (Ax B)(x2 x 1) (Cx D)(x2 – x 1)Comparing the co-efficient of like powers of x, we haveCo-efficient of x3:A C 0 .(i)Co-efficient of x2:A B – C D 1 . (ii)Co-efficient of x:A B C – D 0 . (iii)ConstantB D 1 . (iv)Subtract (iv) from (ii) we haveA–C 0 (v)A C (vi)Adding (i) and (v), we haveA 0Putting A 0 in (vi), we haveC 0Putting the value of A and C in (iii), we haveB–D 0 (vii)Adding (iv) and (vii)2B 1 B from (vii) B D, thereforeD 12Hence the required partial fraction are12
Chapter 496Partial Fraction110x 2 222(x x 1) (x x 1)11 2(x 2 x 1) 2(x 2 x 1)0x i.e.,Exercise 4.3Resolve into partial fraction:Q.1Q.3Q.5Q.7Q.9Q.11x 2 3x 1(x 2)(x 2 5)3x 7(x 3)(x 2 1)1(x 1)(x 2 1)3x 2 x 1(x 1)(x 2 x 3)x5x4 11x3 1Q.2Q.4Q.6Q.8Q.10Q.12x2 x 2(x 1)(x 2 3)13(x 1)3x 7(x 2 x 1)(x 2 4)x ax (x a)(x 2 a 2 )x2 x 1(x 2 x 2)(x 2 2)2x 2 3x 3(x 2 1)(x 2 4)Answers 4.3Q.113 2x 2 x 5Q.3 Q.51x 1 2(x 1) 2(x 2 1)Q.6Q.7Q.812x 2 2x 1 x x 311 2x 1 x 31(x 2)Q.4 3(x 1) 3(x 2 x 1)Q.2
Chapter 4Q.9Q.10Q.11Q.1297Partial Fraction11x 24(x 1) 4(x 1) 2(x 1)173x 2 3(x 1) 6(x 2) 2(x 2 2)1x 2 3(x 1) 3(x 2 x 1)713x 1 10(x 1) 10(x 1) 5(x 2 4)x 4.11Type IV: Quadratic repeated factorsWhen the denominator has repeated Quadratic factors.Example 1:Resolve into partial fractionx2(1 x)(1 x 2 )2Solution:x2ABx C Dx E (1 x)(1 x 2 )2 1 x (1 x 2 ) (1 x 2 )2Multiplying both sides by L.C.M. i.e., (1 x)(1 x 2 )2 on bothsides, we havex2 A(1 x2)2 (Bx C)(1 – x)(1 x2) (Dx E)(1 – x) (i)x2 A(1 2x2 x4) (Bx C)(1 – x x2 – x3) (Dx E)(1 – x)x 1 in eq. (i), we havePut 1 – x 0 22 2(1) A(1 (1) )1 4A A 14x2 A(1 2x2 x4) B(x – x2 x3 – x4) C(1 – x x2 – x3) D(x–x2) E(1 – x) (ii)Comparing the co-efficients of like powers of x on both sides inEquation (II), we haveCo-efficient of x4:A–B 0 . (i)Co-efficient of x3:B–C 0 . (ii)2Co-efficient of x:2A – B C – D 1 . (iii)Co-efficient of x:B–C D–E 0 . (iv)Co-efficient term:A C E 0 . (v)from (i),B A B 14A 14
Chapter 498Partial Fractionfrom (i) from (iii)B CC 14C 14D 2A – B C – 1 1 1 1 1 4 4 4 2 D from (v)12E –A–CE 1 11 4 42Hence the required partial fractions are by putting the values of A,B, C, D, E,11 111x x 4 4 4 2222 21 x 1 x(1 x )1(x 1)x 1 24(1 x) 4(1 x ) 2(1 x 2 )2Example 2:x2 x 2Resolve into partial fractions 2 2x (x 3)2Solution:Letx 2 x 2 A B Cx D Ex F x 2 (x 2 3)2 x x 2 x 2 3 (x 2 3) 2Multiplying both sides by L.C.M. i.e., x 2 (x 2 3)2 , we havex2 x 2 Ax(x 2 3)2 B(x 2 3)2 (cx D)x 2 (x 2 3) (Ex F)(x 2 )Putting x 0 on both sides, we have2 B (0 3)22B 2 9B 9Now x 2 x 2 Ax(x 4 6x 2 9) B(x 4 6x 2 9) C(x5 3x2 ) D(x4 3x2 ) E(x3 ) Fx2x2 x 2 (A C)x5 (B D)x 4 (6A 3C E)x3
Chapter 499Partial Fraction (6B 3D F)x2 (x 9B)Comparing the co-efficient of like powers of x on both sides of Eq.(I), we haveCo-efficient of x5:A C 0 (i)Co-efficient of x4:B–D 0 (ii)Co-efficient of x3:6A 3C E 0 (iii)Co-efficient of x2:6B 3D F 1 (iv)Co-efficient of x:9A 1 (v)Co-efficient term:9B 1 (vi)from (v)9A 1 A 19from (i) A C 0C –AC –19from (i) B D 0D –BD –29from (iii) 6A 3C E 1 1 6 3 E 0 9 9 3 6E 9 91E 3from (iv) 6B 3D F 1F 1 – 6B – 3D 2 2 1 6 3 9 9
Chapter 4100 1 F Partial Fraction12 6 9 913Hence the required partial fractions are1 2121 1 x x 9 9 99 3 322x xx 3(x 2 3)212x 2x 1 2 9x 9x9(x 2 3) 3(x 2 3)2Exercise 4.4Resolve into Partial Fraction:Q.17(x 1)(x 2 2)2Q.2x2(1 x)(1 x 2 ) 2Q.35x 2 3x 9x(x 2 3) 2Q.44x 4 3x 3 6x 2 5x(x 1)(x 2 x 1) 2Q.52x 4 3x 3 4x(x 1)(x 2 2)2Q.6x 3 15x 2 8x 7(2x 5)(1 x 2 ) 2Q.749(x 2)(x 2 3)2Q.88x 2(1 x 2 )(1 x 2 )2Q.10x2 2(x 2 1)(x 2 4)22Q.9Q.11x 4 x 3 2x 2 7(x 2)(x 2 x 1)214x x2 1Answers 4.4Q.1Q.2Q.3Q.41x 1x 1 24(1 x) 4(1 x ) 2(1 x 2 )21x2x 3 2 2x x 3 (x 3)2
Chapter 4Q.5Q.6Q.7Q.8Q.9Q.10Q.11101Partial Fraction15(x 1) 2(3x 1) 3(x 1) 3(x 2 2) (x 2 1)22x 3x 2 22x 5 1 x(1 x 2 )21x 2 7x 14 2 x 2 x 3 (x 2 3)21124 1 x 1 x 1 x 2 (1 x 2 )212x 31 2 22x 2 (x x 1)x x 1112 2229(x 1) 9(x 4) 3(x 4)2(x 1)(x 1) 2(x 2 x 1) 2(x 2 x 1)SummaryLet N(x) and D(x) 0 be two polynomials. TheN(x)is called aD(x)proper fraction if the degree of N(x) is smaller than the degree of D(x).For example:Alsox 1is a proper fraction.x 5x 62N(x1 )is called an improper fraction of the degree of N(x) isD(x)greater than or equal to the degree of D(x).For example:x5is an improper fraction.x4 1In such problems we divide N(x) by D(x) obtaining a quotient Q(x) and aremainder R(x) whose degree is smaller than that of D(x).ThusN(x)'R(x)R(x)'whereis proper fraction. Q(x) D(x)D(x)D(x)Types of proper fraction into partial fractions.Type 1:Linear and distinct factors in the D(x)
Chapter 4102Partial Fractionx aAB (x a)(x b) x a x bType 2:Linear repeated factors in D(x)x aABx C 222(x a)(x b ) x a x b2Type 3:Quadratic Factors in the D(x)x aABx C 222x a x b2(x a)(x b)Type 4:Quadratic repeated factors in D(x):x aAx B Cx DEx F (x 2 a 2 )(x 2 b2 ) x 2 a 2 x 2 b 2 (x 2 b 2 ) 2
Chapter 4103Partial FractionShort Questions:Write the short answers of the following:Q.1:What is partial fractions?Q.2:Define proper fraction and give example.Q.3:Define improper fraction and given one example:Q.4:2xResolve into partial fractions (x - 2) (x 5)Q.5:1Resolve into partial fractions: x2 - xQ.6:Resolve7x 25(x 3)(x 4) into partial fraction.Q.7:Resolve1x - 1 into partial fraction:Q.8:x2 1Resolve (x 1)(x - 1) into partial fractions.Q.9:Write an identity equation of28 x2(1 - x2)(1 x2)22x 5Q.10: Write an identity equation of x2 5x 6Q.11: Write identity equation ofx-5(x 1)(x2 3)Q.12: Write an identity equation of6x3 5 x2 - 73x2 - 2x - 1(x - 1) (x -2)(x - 3)Q.13: Write an identity equation of (x - 4) (x - 5) ( x - 6)x5Q.14: Write an identity equation of x4 - 12x4 - 3x2 - 4xQ.15: Write an identity equation of (x 1)(x2 2)2
Chapter 4104Partial Fraction1is .(x 1)(x 2)1Q.17. Form of partial fraction ofis .(x 1)2 (x 2)1Q.18. Form of partial fraction of 2is .(x 1)(x 2)1Q.19. Form of partial fraction of 2is .(x 1)(x 4)2Q16.Form of partial fraction of1is .(x 1)(x 2 1)Q.20. Form of partial fraction of3Answers410Q4. 7(x - 2) - 7(x 5)Q5.-11 xx-143Q6. x 3 x 4111Q7. x2 - 1 2(x - 1) - 2(x 1)11Q8. 1 x 1 x - 1ABQ10. x 2 x 3ABCx DEx FQ9. 1 - x 1 x 1 x2 (1 x2)2ABx CQ11. x 1 x2 3Q12. (2x 3) AB x - 1 3x 1Q13.ABCx DQ14. x x - 1 x 1 x2 1Q16.AB x 1 x 21 Q15.ABx CDx E 2x 1x 2(x2 2)2Q17.ABC 2x 1 (x 1)x 2Ax BC Q19.2x 1 x 2ABx CDx EQ20. 2 2(x 1) (x x 1) x 1Q18.ABC 4-4x-5 x-6Ax BCD 2x 1 x 1 (x 1)2
Chapter 4105Partial FractionObjective Type QuestionsQ.11.2.3.Each questions has four possible answers. Choose the correctanswer and encircle it.If the degree of numerator N(x) is equal or greater than the degreeof denominator D(x), then the fraction is:(a)proper(b)improper(c)Neither proper non-improper (d) Both proper and improperIf the degree of numerator is less than the degree of denominator,then the fraction is:(a)Proper(b)Improper(c)Neither proper non-improper (d) Both proper and improperThe fraction(a)(c)4.24The equivalent partial fraction of(a)(c)7.24(b)(d)ImproperNone of these6x 27are:4x 3 9x(b)(d)3None of thesex 3 3x 2 1The number of partial fractions ofare:(x 1)(x 1)(x 2 1)(a)(c)6.ProperBoth proper and improperThe number of partial fractions of(a)(c)5.2x 5is known as:x 5x 62AB x 1 (x 3)2ABC x 1 x 3 (x 3)2The equivalent partial fraction of(a)(c)Ax B Cx D 2x2 1x 3Ax B Cx D1 2 2x 1x 3(b)(d)35x 11is:(x 1)(x 3)2AB (b)x 1 x 3ABx C(d) x 1 (x 3)2x4is:(x 2 1)(x 2 3)Ax BCx 2(b)2x 1 x 3AxBx (d)x2 1 x2 3
Chapter 48.106Partial fraction of(a)(c)Par
Chapter 4 85 Partial Fraction (II) If
Fractions Prerequisite Skill: 3, 4, 5 Prior Math-U-See levels Epsilon Adding Fractions (Lessons 5, 8) Subtracting Fractions (Lesson 5) Multiplying Fractions (Lesson 9) Dividing Fractions (Lesson 10) Simplifying Fractions (Lessons 12, 13) Recording Mixed Numbers as Improper Fractions (Lesson 15) Mixed Numbers (Lessons 17-25)
teacher education (PBTE) for the educational decision-maker and practitioner. unfamiliar with .PBTE and at the same time as 'a handy reference for those experienced in PBTE. The Source Book can be used as a personal reference'or in study groups for a
Part One: Heir of Ash Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16 Chapter 17 Chapter 18 Chapter 19 Chapter 20 Chapter 21 Chapter 22 Chapter 23 Chapter 24 Chapter 25 Chapter 26 Chapter 27 Chapter 28 Chapter 29 Chapter 30 .
Year 5 is the first time children explore improper fractions in depth so we have added a recap step from Year 4 where children add fractions to a total greater than one whole. What is a fraction? Equivalent fractions (1) Equivalent fractions Fractions greater than 1 Improper fractions to mix
Adding & Subtracting fractions 28-30 Multiplying Fractions 31-33 Dividing Fractions 34-37 Converting fractions to decimals 38-40 Using your calculator to add, subtract, multiply, divide, reduce fractions and to change fractions to decimals 41-42 DECIMALS 43 Comparing Decimals to fractions 44-46 Reading & Writing Decimals 47-49
fractions so they have the same denominator. You can use the least common multiple of the denominators of the fractions to rewrite the fractions. Add _8 15 1 _ 6. Write the sum in simplest form. Rewrite the fractions as equivalent fractions. Use the LCM as the denominator of both fractions
(a) Fractions (b) Proper, improper fractions and mixed numbers (c) Conversion of improper fractions to mixed numbers and vice versa (d) Comparing fractions (e) Operations on fractions (f) Order of operations on fractions (g) Word problems involving fractions in real life situations. 42
Decimals to Fractions (Calculator) [MF8.13] Ordering Fractions, Decimals and Percentages 1: Unit Fractions (Non-Calculator) [MF8.14] Ordering Fractions, Decimals and Percentages 2: Non-Unit Fractions (Non-Calculator) [MF8.15] Ordering Fractions, Decimals and Percentages 3: Numbers Less than 1 (Calculator) [MF8.16] Ordering Fractions, Decimals .