Welcome to three examples,
on determining the values for which
a rational expression is undefined.
The main thing to remember,
is that a fraction bar means division,
so if we have the fraction a over b,
this means a divided by b,
and because division by zero is undefined,
to determine the values where
a rational expression is undefined,
we need to determine the values
for which the denominator would be equal to zero.
So looking at our first example,
the rational expression is undefined
when the quantity four x minus 32 is equal to zero.
So, let's solve the equation four x minus 32 equals zero,
first step is to isolate the variable term,
so we add 32 to both sides.
Simplifying, negative 32 plus 32 is zero,
so we have four x equals 32.
Solve for x, we divide both sides by four.
We have x equals eight, which means,
we would have division by zero when x equals eight,
and therefore the expression is undefined,
when x equals eight.
We could have also solved this equation by factoring,
let's show that method as well.
Beginning with the quantity four x minus 32 equals zero,
we factor the left side, the greatest common factor is four,
the factor of four out of the left side,
we'd be left with the quantity x minus eight.
This product is equal to zero,
when the factor of x minus eight equals zero.
Solve for x we'd add eight to both sides,
giving us the same value of x equals eight,
where the expression is undefined.
Looking at our next example, the only difference here
is that we have two binomial factors in the denominator,
so this expression is undefined,
when the quantity x minus 10,
times the quantity x minus four, is equal to zero.
This product equals zero,
when the factor of x minus 10 equals zero or,
when the factor of x minus four is equal to zero.
Solving for x here we add 10 to both sides,
giving us x equals 10.
Solving for x here we add four to both sides,
giving us x equals four.
Which means the rational expression is undefined,
when x equals four, or x equals 10.
Notice how we do not consider the numerator,
when determining the values for which
a rational expression is undefined.
And for our last example, the only difference here
is the denominator is not in factored form.
The rational expression is undefined,
meaning we have division by zero, when the quantity
x squared minus nine x minus 10 is equal to zero.
So the left side factors into two binomial factors,
because the first term is x squared,
we have a factor of x here and x here,
in the first positions of the binomial factors,
and then for the second terms in the binomial factors,
we're looking for the factors
of negative 10 to add to negative nine,
and because negative 10 times one equals negative 10,
and, we have 10 plus one equals negative nine,
the two factors we're looking for,
are negative 10 and positive one.
So one factor would be x plus one,
the second factor would be x minus 10.
Of course the order of the factors don't matter.
This product is equal to zero,
when x plus one equals zero or, when x minus 10 equals zero.
Solving for x here we subtract one on both sides,
giving us x equals negative one.
Solving for x here we add 10 to both sides,
giving us x equals 10.
So these are the two x values,
that make the denominator equal to zero,
and therefore these are the x values,
where the expression is undefined.
So we enter x equals negative one comma positive 10.
I hope you found this helpful.