Partial-Fraction Decompisition

Steven Watt, Lyanne Lebaquin, Wilson Tam

What is Partial-Fraction Decomposition?

• 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.

How do You Decompose a Fraction?

3x+2x2+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

How do You Decompose a Fraction?

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

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

How do You Decompose a Fraction?

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=2B=1

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

Example One

• 3x+1 x2+4x+3

Example Two

• 7x+11x2+2x+2

 

Example Three

•  8x-42      x2+3x-18

Example Four

•     4x2        (x-1)(x-2)