Question

Decompose into partial fractions.

(5x^3 -x^2 + 8x -55) / (x^4 + 5x^3 + 11x^2)

Answer 1

\[\frac{2 x-11}{x^2+5 x+11}-\frac{5}{x^2}+\frac{3}{x} \]

Answer 2

\[\frac{5x^{3}-x^{2}+8x-55}{x^{4} +5x^{3}+11x^{2}} = \frac{5x^{3}-x^{2}+8x-55}{x^{2}(x^{2}+5x+11)} = \frac{A}{x^{2}}+\frac{B}{x}+\frac{Cx+D}{x^{2}+5x+11}\]

Answer 3

Ok so what do i do after that?

Answer 4

I actually dont even know what decmpose into partial fractions mean...i assumed it meant breaking it up to smaller fractions but i'm not sure how to do that.

Is taht the end of the problem?

Answer 5

\[= \frac{A(x^{2}+5x+11) + Bx(x^{2}+5x+11)+(Cx+D)x^{2}}{x^{2}(x^{2}+5x+11}\]

Answer 6

yeah its breaking it into fractions which are equivalent to the original single fraction

Answer 7

Now find values for A,B,C,D so it matches original numerator
\[A(x^{2}+5x+11) + Bx(x^{2}+5x+11)+(Cx+D)x^{2} = 5x^{3} -x^{2}+8x-55\]

Answer 8

\[(B+C)x^{3}+(A+5B+D)x^{2}+(5A+11B)x +11A = 5x^{3}-x^{2}+8x-55\]
\[B+C = 5\]
\[A+5B+D = -1\]
\[5A+11B = 8\]
\[11A = -55\]

Answer 9

A = -5
B = 3
C=2
D = -11

Answer 10

\[= -\frac{5}{x^{2}}+\frac{3}{x}+\frac{2x-11}{x^{2}+5x+11}\]

Answer 11

THANK YOU SOOO MUCH! I'm a visual learner so this was such a great step by step process. I really appreciate your help.

Answer 12

your welcome

Answer 13

I have another question..do u mind? It's just to confirm ansnwer about the domain and range of a function.

Answer 14

yeah sure

Answer 15

Is the domain and range of a function any number that makes it positive if the function is a sqrt root. For example the sqrt root of x^2 +44, \[dom: x \ge \sqrt{44} and range is y \ge 0\]

Answer 16

ok range is correct, range of sqrt is all positive numbers
domain is off, since x^2 +44 is always greater than zero, x is all real numbers

Answer 17

but doesnt real #'s include neg. #'s and a sqt can't be negative

Answer 18

that is true, but here it applies to x^2+44 not just x
x could be anything

Answer 19

oh i see now... so if it were like \[f(x) = \sqrt{x ^{2}-44}\]
then domain would be x\[x \ge -2\]

Answer 20

i am wrong about the range, it should be y>= sqrt(44)

it helps if you can graph the function to see domain and range

Answer 21

is my previous note correct with teh domain?

Answer 22

not quite
x^2 -44 > 0
x^2 > 44
x > sqrt(44) and x< -sqrt(44)

Answer 23

hope this helps
got to go

Answer 24

THANKS ALOT ! =) IT REALLY DID