Skip to content
Exercise 11.5 · Q20

Q.x3(x−1)(x−2)\dfrac{x^{3}}{(x-1)(x-2)}

Tamil Nadu DgeTextbookSubjectiveImportance★★★★★
31% · 40/129 Questions
🔒 Locked · start free trial →

You're viewing a preview — the full solution, concept, methods & PYQ mapping are locked.

Start your 14-day free trial to unlock the full solution →

This is an improper fraction (deg⁡\deg numerator >deg⁡>\deg denominator), so long division comes first, and partial fractions are applied only to the proper remainder.

Step 1. Long-divide. (x−1)(x−2)=x2−3x+2(x-1)(x-2)=x^2-3x+2. Dividing x3x^3 by x2−3x+2x^2-3x+2: x3=(x2−3x+2)(x+3)+(7x−6)x^3=(x^2-3x+2)(x+3)+(7x-6), so x3(x−1)(x−2)=x+3+7x−6(x−1)(x−2)\dfrac{x^3}{(x-1)(x-2)}=x+3+\dfrac{7x-6}{(x-1)(x-2)}.

Step 2. Partial-fraction the proper remainder. 7x−6(x−1)(x−2)=Ax−1+Bx−2\dfrac{7x-6}{(x-1)(x-2)}=\dfrac{A}{x-1}+\dfrac{B}{x-2}, so 7x−6=A(x−2)+B(x−1)7x-6=A(x-2)+B(x-1).

Step 3. Solve for A,BA,B. At x=1x=1: 1=A(−1)⇒A=−11=A(-1)\Rightarrow A=-1. At x=2x=2: 8=B(1)⇒B=88=B(1)\Rightarrow B=8. …

Start your 14-day free trial to unlock the full solution →

Unlock everything free for 14 days

  • Full step-by-step solutions
  • Concept-first explanations
  • Methods, shortcuts & mistakes
  • PYQ mapping + timed mock tests
Start 14-day free trial →See Plans & Pricing →

Full access for 14 days. No credit card required.

←Previous question · 11.7.2Q19 — dfrac3x-9(x-1)(x+2)(x²+1)→Next section · 11.7.3Method of Substitution or Change of Variable