Write the partial fraction decomposition of the rational expression.

Question

Quizplus · Accepted Answer

Since the given expression has a denominator of degree 2, we can rewrite it in the form of partial fraction decomposition as:

$$\frac{11eb11e8\_19a9\_4305\_ad38\_056589d84214}{(a+bi)(a-bi)}=\frac{A}{a+bi}+\frac{B}{a-bi}$$

where A and B are constants to be determined.

Multiplying both sides by the common denominator, we get:

$$11eb11e8\_19a9\_4305\_ad38\_056589d84214 =A(a-bi)+B(a+bi)$$

Setting $a=bi$, we get:

$$11eb11e8\_19a9\_4305\_ad38\_056589d84214 =B(2bi),$$

which implies that $B=\frac{11eb11e8\_19a9\_4305\_ad38\_056589d84214}{2bi}.$

Similarly, setting $a=-bi$, we get:

$$11eb11e8\_19a9\_4305\_ad38\_056589d84214 =A(-2bi),$$

which implies that $A=-\frac{11eb11e8\_19a9\_4305\_ad38\_056589d84214}{2bi}.$

Therefore, the partial fraction decomposition of the given expression is:

$$\frac{11eb11e8\_19a9\_4305\_ad38\_056589d84214}{(a+bi)(a-bi)}=-\frac{11eb11e8\_19a9\_4305\_ad38\_056589d84214}{2bi(a+bi)}+\frac{11eb11e8\_19a9\_4305\_ad38\_056589d84214}{2bi(a-bi)}$$
which simplifies to 
$$\frac{11eb11e8\_19a9\_4305\_ad38\_056589d84214}{(a+bi)(a-bi)}=\frac{11eb11e8\_19a9\_4309\_ad38\_5bd353267990}{a+bi}-\frac{11eb11e8\_19a9\_4309\_ad38\_5bd353267990}{a-bi}$$
So, the answer is D.

Write the Partial Fraction Decomposition of the Rational Expression