Write an informal explanation of partial fraction decomposition.

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Malena · Accepted Answer

Step 1Partial-fraction decomposition is the process of starting with the simplified answer and taking it back apart, of &quot;decomposing&quot; the final expression into its initial polynomial fractions.For example: 2x+1x+1=2(x+1)x(x+1)+1(x)(x+1)(x)=2x+xx(x+1)+xx(x+1)=2x+2+xx(x+1)=3x+2x(x+1)=3x+2x2+xStep 2To decompose a fraction, you first factor the denominator. Let&#039;s work backwards from the example above. The denominator is x2+x which factors as x(x+1).Step 3Then you write the fractions with one of the factors for each of the denominators. Of course, you don&#039;t know what the numerators are yet, so you assign variables (usually capital letters) for these unknown values:Ax+Bx+1Step 4Then you set this sum equal to the simplified result:3x+2x(x+1)=Ax+Bx+1Step 5Multiply through by the common denominator of x(x + 1) gets rid of all of the denominators:[3x+2x(x+1)][x(x+1)1]=[Ax][x(x+1)1]+[Bx+1][x(x+1)1]3x+2=A(x−1)−B(x)Multiply things out and group the x-terms and the constant terms3x+2=Ax+A1+Bx3x+2=(A+B)x+(A)1(3)x+(2)1=(A+B)x+(A)1Step 6For the two sides to be equal, the coefficients of the two polynomials must be equal. So you &quot;equate the coefficients&quot; to get:3=A+B2=AStep 7This creates a system of equations that you can solve:A=2B=1Step 8Then the original fractions were (as we already know) the following:3x+2x2+x=2x+1x+1

Write an informal explanation of partial fraction decomposition.

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