Write the partial fraction decomposition of the rational expression.Check your result algebraically. $\frac { 4 x ^ { 2 } + 3 x + 20 } { \left( x ^ { 2 } + 5 \right) ^ { 2 } }$
A) $\frac { 4 } { x ^ { 2 } + 5 } + \frac { 3 } { \left( x ^ { 2 } + 5 \right) ^ { 2 } }$
B) $\frac { 4 x + 20 } { x ^ { 2 } + 5 } + \frac { 3 x } { \left( x ^ { 2 } + 5 \right) ^ { 2 } }$
C) $\frac { 3 x + 4 } { \left( x ^ { 2 } + 5 \right) ^ { 2 } }$
D) $\frac { 4 } { x ^ { 2 } + 5 } - \frac { 3 x } { \left( x ^ { 2 } + 5 \right) ^ { 2 } }$
E) $\frac { 4 } { x ^ { 2 } + 5 } + \frac { 3 x } { \left( x ^ { 2 } + 5 \right) ^ { 2 } }$

Question

Write the partial fraction decomposition of the rational expression.Check your result algebraically.​ $\frac { 4 x ^ { 2 } + 3 x + 20 } { \left( x ^ { 2 } + 5 \right) ^ { 2 } }$ ​
A) $\frac { 4 } { x ^ { 2 } + 5 } + \frac { 3 } { \left( x ^ { 2 } + 5 \right) ^ { 2 } }$
B) $\frac { 4 x + 20 } { x ^ { 2 } + 5 } + \frac { 3 x } { \left( x ^ { 2 } + 5 \right) ^ { 2 } }$
C) $\frac { 3 x + 4 } { \left( x ^ { 2 } + 5 \right) ^ { 2 } }$
D) $\frac { 4 } { x ^ { 2 } + 5 } - \frac { 3 x } { \left( x ^ { 2 } + 5 \right) ^ { 2 } }$
E) $\frac { 4 } { x ^ { 2 } + 5 } + \frac { 3 x } { \left( x ^ { 2 } + 5 \right) ^ { 2 } }$

Quizplus · Accepted Answer

The correct partial fraction decomposition of the given rational expression is $\frac { 4 } { x ^ { 2 } + 5 } + \frac { 3 x } { \left( x ^ { 2 } + 5 \right) ^ { 2 } }$. This is because the denominator is a square of a binomial, and the numerator is a polynomial of degree less than 4 (the total degree of the denominator). The decomposition must account for both the linear term in the numerator and the constant term, which matches the structure of option E.

Write the Partial Fraction Decomposition of the Rational Expression 4x2+3x+20(x2+5)2\frac { 4 x ^ { 2 } + 3 x + 20 } { \left( x ^ { 2 } + 5 \right) ^ { 2 } }(x2+5)24x2+3x+20​

Write the Partial Fraction Decomposition of the Rational Expression $\frac { 4 x ^ { 2 } + 3 x + 20 } { \left( x ^ { 2 } + 5 \right) ^ { 2 } }$