Question

find the domain and simplify

Answer 1

\[\frac{ x+1 }{ 3x-3 } - \frac{ 4 }{ x^2-3x+2 }\]

Answer 2

we have to factorize both denominators

Answer 3

i know and i did the first but dont know how to do the decond
first one is 3(x-1)

Answer 4

for the second one, you have to search for two numbers such that their sum is 3, and their product is 2

Answer 5

1 and 2

Answer 6

correct!

Answer 7

so we can write this:
\[\Large {x^2} - 3x + 2 = \left( {x - 1} \right)\left( {x - 2} \right)\]

Answer 8

now we have to find the domain

Answer 9

therefore:
\[\Large \begin{gathered}
\frac{{x + 1}}{{3x - 3}} - \frac{4}{{{x^2} - 3x + 2}} = \hfill \\
\hfill \\
= \frac{{x + 1}}{{3\left( {x - 1} \right)}} - \frac{4}{{\left( {x - 1} \right)\left( {x - 2} \right)}} \hfill \\
\end{gathered} \]

Answer 10

yes! we have to find the domain

Answer 11

in order to do that, we note that we can not divide by zero

Answer 12

Domain is all real numbers except when x = 1 or 2

Answer 13

is that right?

Answer 14

so we have to request that:
\[\Large \begin{gathered}
x - 1 \ne 0 \hfill \\
\left( {x - 1} \right)\left( {x - 2} \right) \ne 0 \hfill \\
\end{gathered} \]
yes! correct!

Answer 15

ok now we have to find the common denominator

Answer 16

right! we have to find the common denominator between the two denominators:

\[\Large \begin{gathered}
3\left( {x - 1} \right) \hfill \\
\left( {x - 1} \right)\left( {x - 2} \right) \hfill \\
\end{gathered} \]

Answer 17

3?

Answer 18

no, we have to take all uncommon and common factors with the highest exponent

Answer 19

ok so how do we do that

Answer 20

it is simple, x-1 is a common factor, whereas 3 and x-2 are uncommon factor, so the common denominator is:
3*(x-1)*(x-2)

Answer 21

and hence, we can write this:
\[\Large \begin{gathered}
\frac{{x + 1}}{{3x - 3}} - \frac{4}{{{x^2} - 3x + 2}} = \hfill \\
\hfill \\
= \frac{{x + 1}}{{3\left( {x - 1} \right)}} - \frac{4}{{\left( {x - 1} \right)\left( {x - 2} \right)}} = \hfill \\
\hfill \\
= \frac{{...}}{{3\left( {x - 1} \right)\left( {x - 2} \right)}} \hfill \\
\end{gathered} \]

Answer 22

so what is the common denominator

Answer 23

it is:
\[\Large {3\left( {x - 1} \right)\left( {x - 2} \right)}\]

Answer 24

now we have to divide the common denominator by the first numerator, namely what is:
\[\Large \frac{{3\left( {x - 1} \right)\left( {x - 2} \right)}}{{3\left( {x - 1} \right)}} = ...?\]

Answer 25

you can cancel out common factors