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Partial-Fraction Decompisition

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Partial-Fraction Decompisition

Steven Watt, LyanneLebaquin, Wilson Tam

- Partial-fraction decomposition is the process of starting with the simplified answer and taking it back apart, of “decomposing” the final expression into its initial polynomial fractions.

3x+2

x2+2

Step 1: First, factor the denominator. The denominator in the example above is x2+x, which factors as x(x+1)

Step 2: Write the fractions with one of the factors for each of the denominators. Since you don’t know what the numerators are yet, assign variables for the unknown values.

Step 3: Next set this sum equal to the simplified result

Step 4: Multiply through by the common denominator of x(x+1)

to get rid of all the denominators.

Which will leave you with:

3x+2=A(x+1)+B(x)

Step 5: Multiply things out, and group the x-terms and the constant terms.

3x + 2 = Ax + A1 + Bx3x + 2 = (A + B)x + (A)1 (3)x + (2)1 = (A + B)x + (A)1

Step 6: For the two sides to be equal, the coefficients of the two polynomials must be equal. So you make the coefficients equal and get:

3 = A + B

2 = A

Step 7: So we can tell that:

A=2

B=1

Step 8: Plug in the now known values for A and B into

- 3x+1 x2+4x+3

- 7x+11x2+2x+2

- 8x-42
x2+3x-18

- 4x2
(x-1)(x-2)