Question

3x^2+5x/(x-1)(x-2)^2=dx​

Answer 1

Answer:

Answer

0

3x−2(x+1)2(x−3)2=A(x+1)2+Bx+1+C(x−3)2+Dx−3

Multiply buy the least common denominator ( (x+1)2(x−3)2 ) to eliminate the fractions.

3x−2=A(x−3)2+B(x+1)(x−3)2+C(x+1)2+D(x+1)2(x−3)

Multiply the binomials (sorry about the long equations)

3x−2=A(x2−6x+9)+B(x3−5x2+3x+9)+...

...C(x2+2x+1)+D(x3−x2−5x−3)

Distribute A, B, C, and D:

3x−2=Ax2−6Ax+9A+Bx3−5Bx2+3Bx+9B+...

...Cx2+2Cx+C+Dx3−Dx2−5Dx−3D

If two polynomials are equal, then they have to have the same number of like terms. Create four equations for the different powers of x.

x3 : 0=B+D Equation 1

x2 : 0=A−5B+C−D Equation 2

x1 : 3=−6A+3B+2C−5D Equation 3

x0 : −2=9A+9B+C−3D Equation 4

I choose to solve by elimination:

Use equation 1 to eliminate the D’s in the other equations.

Eq2 + Eq1: 0=A−4B+C Eq5

Eq3 + 5Eq1: 3=−6A+8B+2C Eq6

Eq4 +3Eq1: −2=9A+12B+C Eq7

Now use Equation 5 to eliminate the C’s from equations 6 and 7

Eq6 - 2Eq5: 3=−8A+16B Eq8

Eq7 - Eq5: −2=8A+16B Eq9

Step-by-step explanation:

hope it's helpful