Decompose the fraction into partial fractions. a) x2(x−1)3=? b) x2−7x+3x3−2x2+x=? c) 4x2+7x+4}{x(x+1)2}=? d) 4x2+x+6x3+2x=?

Step 1 x2(x−1)2 x2(x−1)3=Ax−1+B(x−1)2+C(x−1)3 Multiply equatim by denominator x2(x−1)3(x−1)3=A(x−1)3x−1+B(x−1)3(x−1)2+C(x−1)3(x−1)3 ⇒x2=A(x−1)2+B(x−1)+C for the denominator root 1. Put x=1 into the equation. 12=A(1−1)2+B(1−1)+C ⇒C=1 Put the value of c in the equation to know the un known. x2=A(x−1)2+B(x−1)+1 ⇒x2=Ax2−2Ax+A+Bx=B+1 ⇒x2(A−1)+x(−2A+B)+1(A−B+10)=0 ⇒A−1=0, −2+B=0, A−B+1=0 ⇒A=1, +2A=B, 1−2+1=0 ⇒B=2 Hence portial fraction is x2(x−1)3=1x+1+2(x−1)2+1(x−1)3 Step 2 x2−7x+3x3−2x2+x=1 Factor of x3−2x2+x=x(x2−

"Decompose the fraction into partial fractions. a) x2(x−1)3=?..." solved and explained

Dolly Robinson

Answered question

2021-10-10

Decompose the fraction into partial fractions.
a)

\frac{x^{2}}{{(x - 1)}^{3}} = ?

\frac{x^{2} - 7 x + 3}{x^{3} - 2 x^{2} + x} = ?

\frac{4 x^{2}}{+} 7 x + 4} {x {(x + 1)}^{2}} = ?

\frac{4 x^{2} + x + 6}{x^{3} + 2 x} = ?

Answer & Explanation

Leonard Stokes

Skilled2021-10-11Added 98 answers

Step 1

\frac{x^{2}}{{(x - 1)}^{2}}

\frac{x^{2}}{{(x - 1)}^{3}} = \frac{A}{x - 1} + \frac{B}{{(x - 1)}^{2}} + \frac{C}{{(x - 1)}^{3}}

Multiply equatim by denominator

\frac{x^{2} {(x - 1)}^{3}}{{(x - 1)}^{3}} = \frac{A {(x - 1)}^{3}}{x - 1} + \frac{B {(x - 1)}^{3}}{{(x - 1)}^{2}} + \frac{C {(x - 1)}^{3}}{{(x - 1)}^{3}}

\Rightarrow x^{2} = A {(x - 1)}^{2} + B (x - 1) + C

for the denominator root 1.
Put

x = 1

into the equation.

1^{2} = A {(1 - 1)}^{2} + B (1 - 1) + C

\Rightarrow C = 1

Put the value of c in the equation to know the un known.

x^{2} = A {(x - 1)}^{2} + B (x - 1) + 1

\Rightarrow x^{2} = A x^{2} - 2 A x + A + B x = B + 1

\Rightarrow x^{2} (A - 1) + x (- 2 A + B) + 1 (A - B + 10) = 0

\Rightarrow A - 1 = 0, - 2 + B = 0, A - B + 1 = 0

\Rightarrow A = 1, + 2 A = B, 1 - 2 + 1 = 0

\Rightarrow B = 2

Hence portial fraction is

\frac{x^{2}}{{(x - 1)}^{3}} = \frac{1}{x + 1} + \frac{2}{{(x - 1)}^{2}} + \frac{1}{{(x - 1)}^{3}}

Step 2

\frac{x^{2} - 7 x + 3}{x^{3} - 2 x^{2} + x} = 1

Factor of

x^{3} - 2 x^{2} + x = x (x^{2} -

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