How to factor and use the zero product property
Zero Product Property
Timecodes:
in this lesson we're going to focus on
the zero product property
and the basic idea behind the zero
product property
is
if you're multiplying two numbers a and
b
and if a and b
equals zero
then either a or b must be zero
because zero times anything
is zero
now let's think about this let's say we
have two numbers that multiply
to zero
the only way this could happen is if one
of those numbers is zero
for instance if we have eight
if the first number is eight the second
number has to be zero
it really doesn't matter what the first
number is if it's 6 12 15 as long as the
second number is zero the whole thing is
going to be zero
if the first number is zero it doesn't
matter what the second number is it
could be negative four seven
twelve
0 times anything is zero
so whenever you have two things that
multiply to zero
one of those things
has to be zero
now this property is very useful
when solving quadratic equations
typically
when you factor a trinomial
you might get something that looks like
this
and you need to solve for x
using a zero product property we can do
that
if either x minus three or x plus two is
equal to zero
this entire expression will be equal to
zero
so using the zero product property
we can break this single equation into
two parts
we can set x minus three equal to zero
and we can set x plus two equal to zero
because if just one of these
two factors equals zero the whole thing
is going to be zero
and so that's why we can do this it's
because of the zero product property
now once we have these two equations we
can go ahead and solve it
to get the answer for the first one we
just got to add 3 to both sides
and we'll get x is equal to positive 3.
for the second one
we need to subtract both sides by 2
and we'll get x is equal to negative 2.
so that's one of the applications
of using
the zero product property
it's very useful when solving quadratic
equations especially when you're
factoring
for the sake of practice
go ahead and calculate the value of x
for these two equations
the first one is going to be 3x
times x minus seven is equal to zero
and for the second one
it's going to be two x minus three
times three x minus five equal to zero
so use the zero product property
to calculate the value of x for each of
these two equations
now for the first one
what we can do is we can set each factor
equal to zero
so we can set 3x equal to zero
and x minus seven equal to zero
for the first one we need to divide both
sides by three
three x divided by three is simply x
zero divided by three is zero
so the first answer is just x is equal
to zero
for the second equation
we simply need to add seven
to both sides
and we get the answer x is equal to
seven
so if we were to plug in 0
or 7
into the original equation
it's going to work
for instance if we plug in 0
into each x value we'll have three times
zero
times zero minus seven
three times zero is zero
zero minus seven is negative seven
zero times negative seven is zero
so this works
now if we plug in seven
it will work as well
three times seven
and then seven minus seven that's gonna
be zero three times seven is twenty one
seven minus 7 is 0
21 times 0 is 0.
so using the zero product property
we can solve for x
whenever it's in factored form
now let's try the other example
now we're going to follow the same
process
we're going to set each factor equal to
zero
so we'll break it up into two equations
two x minus three is equal to zero
and three x minus five is equal to zero
so let's begin by adding three
to both sides
so we'll get 2x is equal to 3
and then we'll divide both sides by 2.
so the first answer that we get
is x is equal to three over two
if we were to plug this in to the
original equation
and this part will equal zero which
means the whole equation will equal zero
now for the second one we're going to
add five
to both sides
so we'll get 3x is equal to 5
and then we'll divide both sides by
3.
so we'll get x is equal to 5 over 3.
so if you were to plug in any one of
these two x values into the original
equation
the whole thing is going to equal 0.
so that's how you can solve equations
using the zero product property
first it needs to be in factored form
like this
and then you could set each factor equal
to zero
and then solve for x
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