Welcome to Georgia Highlands College Math 97 and Math 99 video tutorials. In
this video segment, we’ll answer the question, what is the zero product
principle?
Well the zero product principle is a very simple principle yet very powerful
when it comes to solving polynomial equations. It simply states that if I'm
multiplying factors together, so if I'm creating a product A times B, and that
product is equal to 0, the
only way for that to occur is if either A is zero or B is zero.
So simply stated, if you're multiplying two things together that make zero, then
you
had to be multiplying with zero. So let's take a closer look at the property
itself and a couple of examples.
Okay, so at the top of the screen here, I just have stated the zero product
principal in phrased words here. So if the product of two algebraic expressions
is zero, then at least one of the factors had to be zero.
In other words, if A times B equals 0, then A equals 0 or B equals 0.
So let’s take a look at this example here. I have two factors being multiplied
equaling 0.
I have a factor of X plus 3. I have a factor of X minus 4. And you might
recognize that it
follows that same sort of form as something being multiplied with something else
equaling 0.
So we’re set to use our zero product principal.
Now to apply this principle, we’re going to state that one of these factors had
to at least be 0.
So I simply write that out as the factor X plus 3 either had to equal zero, for
me to be able to multiply these two things together to make zero, or X minus 4
had to equal 0.
So right here I'm applying the zero product principal by setting each of those
factors equal to zero.
Then it’s easy from here. You simply solve for X in each of these equations.
So here I would subtract three from both sides and find that X is equal to
negative 3 and in the other equation, I would add 4 to both sides and get that X
is equal to 4.
Now we’re talking about an equation here, so when you find solutions to an
equation you should be able to plug them back into the original equation and
have it equal out with a true statement.
So let's check each of these, negative 3 and 4, and see that they are actually
solutions to the equation X plus 3 times X minus 4 equals 0.
So I'll just rewrite my equation to begin with before I plug-in my value.
And in this first one I'm going to plug-in the value negative 3 wherever I see
X. So I have negative 3 plus 3 times negative 3 minus 4 and I’m going to see if
that equals 0.
Negative 3 plus 3 is 0, negative 3 minus 4 is negative 7, and 0 times anything
is just 0. So sure enough, I have a true statement there, meaning that X equals
negative 3 is a solution to my equation, and now checking positive 4 by plugging
in.
So I'll start by just rewriting my original equation X plus 3 times X minus 4
equals 0, and next I'll plug 4 in for X in both of the factors, and that gives
me 4 plus 3, giving 7, and 4 minus 4 which is 0, and certainly 7 times 0 is 0,
making positive for a solution as well.
So we have verified that using our zero product principal, we find two solutions
to this equation, negative 3 and positive 4, which can either be written as X =
negative 3, 4
or you can write them as a set, negative 3 and 4.
Sometimes it's difficult to draw those braces like that. A lot of my students
just draw squigglies and I tell them I know what that means.
There's one other thing I'd like to point out before we end this video segment,
and that is the fact that this particular equation came from the polynomial
equation X squared minus X minus 12 equals 0. So we have that standard form and
I just got that by multiplying X plus 3 times X minus 4.
So you can see that that equation actually started out as a quadratic equation
in standard form. Just to point out my A would be positive 1 here, my B would be
negative 1, and my C is negative 12. So this should give you a little bit of
insight into what we’re about to move into, which is solving quadratic equations
We use factoring in applying the zero product principle to do so.
I hope that this has been helpful for you in understanding the zero product
principal. Once again, it’s a simple principle, but it’s very powerful. All it
states is that if you're multiplying factors together and equaling zero, then
one of those factors had to be zero, and we use this in solving quadratic
equations by setting each of those linear factors up to equaling zero and then
solving for it.
If you have any other questions about the zero product principle, contact your
Highlands instructor.
Thank you.
