Question

write the partial fraction decomposition of the rational expression. Check your result algebraically.

8/ x^2-4x-21

Answer 1

You already have it.

\(\dfrac{8}{x^{2}} - 4x - 21\)

Of course, if you meant \(\dfrac{8}{x^{2} - 4x - 21}\), which you did NOT write, you should first factor the denominator.

Answer 2

oh, sorry/ I meant the second one you had wrote.

Answer 3

Sweet. Remember your Order of Operations. Use parentheses as necessary to clarify intent. Factor away!

Answer 4

did you get
\[8=A(x-7)+B(x+3)\]?

Answer 5

i can show you a real snappy way to do this if you like

Answer 6

start with
\[\frac{8}{(x-7)(x+3)}=\frac{A}{x-7}+\frac{B}{x+3}\] now lets find \(A\) quickly

Answer 7

Sure! I would like to find shortcuts! ^_^

Answer 8

the denominator is of \(A\) is \(x-7\) and that would be zero if \(x=7\) so on the left we are going to cover up the factor of \(x-7\) with your finger, and replace \(x\) by \(7\) i.e.
\[\frac{8}{\cancel{x-7}(7+3)}=\frac{8}{10}=\frac{4}{5}=A\]

Answer 9

similarly to find \(B\) the denominator is \(x+3\) which is zero if \(x=-3\) so cover up that factor on the left, replace \(x\) by \(-3\) and get
\[\frac{8}{(-3+7)\cancel{x+3}}=\frac{8}{-10}=-\frac{4}{5}=B\]

Answer 10

i made a typo there, should have been
\[\frac{8}{(-3-7)\cancel{x+3}}=\frac{8}{-10}=-\frac{4}{5}=B\]

Answer 11

Thank you so much! You are a helpful and brilliant person.

Answer 12

lol
yw