Partial Fraction Decomposition Sec. 7.4a

First, remind me……………………..…what’s a rational function? with In this section, we will write a rational function as a sum of rational functions where each denominator is a power of a linear factor or a power of an irreducible quadratic factor. Example: Each fraction in the sum is a partial fraction, and the sum is a partial fraction decomposition of the original rational function.

Steps to Partial Fraction Decomposition of f(x)/d(x) 1. If the degree of f > degree of d: Divide f by d to obtain the quotient q and the remainder r and write 2. Factor d(x) into a product of factors of the form or , where is irreducible

Steps to Partial Fraction Decomposition of f(x)/d(x) 3. For each factor : The partial fraction decomposition of r(x)/d(x) must include the sum where are real numbers 4. For each factor : The partial fraction decomp. of r(x)/d(x) must include the sum where and are real numbers

Guided Practice Write the terms for the partial fraction decomposition of the given rational function. Do not solve for the corresponding constants. + + Today, we’ll just focus on the linear factors, like these…

by multiplying everything Guided Practice Find the partial fraction decomposition of the given function. Factor the denominator! Write the partial fractions! “Clear the fractions” by multiplying everything by the denominator!

Can we verify this answer algebraically? Graphically? Guided Practice Find the partial fraction decomposition of the given function. Equate the coefficients from each side of the equation! Solve the system! (I don’t care how!!!) Can we verify this answer algebraically? Graphically?

Guided Practice Find the partial fraction decomposition of the given function. Another (easier?) way to solve for the constants: Plug in 5 for x, then plug in –3 for x:

Expand and combine like terms: Guided Practice Find the partial fraction decomposition of the given function. Clear fractions: Expand and combine like terms:

Compare coefficients: Guided Practice Find the partial fraction decomposition of the given function. Compare coefficients: Solve the system:

Guided Practice Find the partial fraction decomposition of the given function. The other way to solve for the A’s: Use x = 2, solve for A 3 Use x = 0, solve for A 1 Use any other x, solve for A 2

More PFD: Denominators with Irreducible Quadratic Factors

Now let’s apply a similar process when working with irreducible quadratic factors… (see “Step 4” in your notes from the previous slides!!)

Find the partial fraction decomposition of Factor the denominator by grouping: Clear fractions: Expand and combine like terms:

Find the partial fraction decomposition of Compare coefficients: Solve the system:

Find the partial fraction decomposition of Clear fractions: Expand and combine like terms:

Find the partial fraction decomposition of Compare coefficients:

Find the partial fraction decomposition of Clear fractions: Expand and combine like terms:

Find the partial fraction decomposition of Compare coefficients: