We are asked to simplify
the given rational expressions.
To simplify a rational expression,
we factor the numerator and denominator,
and then simplify out any common factors.
Looking at the first expression,
notice how we have seven x plus six in the numerator,
the first step in factoring
is always to factor out the greatest common factor.
The greatest common factor between seven x and 63 is seven,
and therefore we factor seven from the numerator.
We factor seven from seven x plus 63,
we're left with x plus nine.
Of course, we can check this by distributing,
we still have seven x plus 63.
Now let's factor the denominator.
Notice how the greatest common factor
among all three terms is one,
and therefore if this trinomial does factor,
it will factor into two binomial factors.
Because the first term is x squared,
what is equal to x times x.
We have an x in the first position of both binomials,
the second terms of the binomial factors
will be the factors of negative 54
that add to positive three.
Well, negative six times positive nine
is equal to negative 54
and so is negative nine times positive six.
Let's check the sum of these factors.
Well, negative six plus nine is equal to positive three,
which means the two factors we need
are negative six and positive nine,
and therefore we can stop.
And because the factors are negative six and positive nine,
one binomial factor is x minus six,
the other binomial factor is x plus nine.
And now before we simplify,
we should make a note of the excluded values,
which in this case are the values of x
that make the denominator equal to zero.
The denominator cannot equal zero because
division by zero is undefined.
Well, notice x minus six is equal to zero of x equal six,
and therefore x can't equal six,
and x plus nine is equal to zero when x equals negative nine
and therefore x also can't equal negative nine.
These are called the excluded values
because if x takes on these values,
the original expression is undefined
because we have division by zero.
And now to simplify the expression,
the quantity x plus nine
divided by the quantity x plus nine simplifies to one,
and therefore the given expression simplifies
to seven divided by the quantity x minus six.
Let's look at the second expression.
Again, the first step in factoring
is to factor out the greatest common factor.
The greatest common factor
of two x squared minus eight is two.
We begin by factoring two from the numerator.
We factor two from two x squared minus eight,
we're left with x squared minus four.
And again, we can check this by distributing.
We still have two x squared minus eight.
We should recognize we can factor this further though,
because now, we have a difference of squares.
Let's look at the denominator first.
Again, the first step in factoring
is to factor out the greatest common factor.
The greatest common factor among all three terms is two
so let's factor two from the denominator.
If we factor two from the denominator,
we're left with the quantity x squared minus x minus six.
And again, we can check this by distributing,
we still have two x squared minus two x minus 12.
Now let's factor both the numerator and denominator further.
In the numerator we have two
times the quantity x squared minus four,
which will factor the two binomial factors
because we have a difference of squares
because the first term is x squared,
we have a factor of x in the first position,
so both binomials,
and because two squared is equal to four,
one binomial factor is x plus two,
the other binomial factor is x minus two.
Now looking at the denominator,
we still have a factor of two,
and then if the trinomial factors,
it will factor the two binomial factors.
Again, because the first term is x squared
which is equal to x times x.
We have an x in the first position.
They're both binomials.
The second terms of the binomial factors
are the factors that have negative six
that add to negative one.
Let's list some of the factors of negative six.
We have negative two times positive three,
which is negative six,
as well as negative three times positive two,
which is equal to negative six.
And now sum of the factors,
negative two plus three is equal to positive one
not negative one,
but negative three plus positive two
is negative one,
and therefore we need the factors of negative three
and positive two.
Which means one binomial factor is x minus three,
the other binomial factor is x plus two.
And again before simplifying,
let's list the excluded values.
Again, the excluded values are the values of x
that make the denominator equal to zero.
X minus three is equal to zero
when x equals three and therefore, x can't equal three,
otherwise we would have division by zero which is undefined.
X plus two is equal to zero when x is equal to negative two,
we must also exclude negative two.
Now that we've listed the excluded values,
let's simplify the rational expression.
The quantity x plus two
divided by the quantity x plus two simplifies to one,
and two divided by two also simplifies to one,
the remaining factors
give us the simplified rational expression.
The rational expression simplifies
to the quantity x minus two,
divided by the quantity x minus three.
I hope you found this helpful.