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Partial-Fraction Decompisition Steven Watt, Lyanne Lebaquin, Wilson Tam

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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.

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How do You Decompose a Fraction? 3x+2 x 2 +2 Step 1: First, factor the denominator. The denominator in the example above is x 2 +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

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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

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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=2 B=1 Step 8: Plug in the now known values for A and B into

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Example One 3x+1 x 2 +4x+3

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Example Two 7x+11 x 2 +2x+2

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Example Three 8x-42 x 2 +3x-18

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Example Four 4x 2 (x-1)(x-2)

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1 Example 1 Evaluate Solution Since the degree 2 of the numerator equals the degree of the denominator, we must begin with a long division: Thus Observe.

1 Example 1 Evaluate Solution Since the degree 2 of the numerator equals the degree of the denominator, we must begin with a long division: Thus Observe.

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