In this video are talking about how to use the Zero Theorem to find solutions or roots to a trinomial
polynomial.
So when we say trinomial remember Try means threes we're talking about three terms in a polynomial equation
like this one.
Whenever we see a trinomial the first thing we want to think about is can we factor it.
So in this particular example we have x squared plus five x plus six equals zero.
We want to see if we can try to factor the left hand side.
So we're going to set up our parentheses like this.
And we want to figure out the factors of x squared so what can we multiply together to get x squared.
Well that's just going to be X and X and now what can we multiply together you get six what we've got
a couple of different options.
We have six in one.
We have two and three.
But the only factors that we can multiply together to get six that we can also add together to get 5
are 3 and 2.
So we're going to say X plus three times x plus two and we can always double check ourselves.
If we multiply this back out we say x times X gives us x squared we say x times 2 gives us a positive
2 x 3 times X gives us a positive 3 x and 3 times 2 gives us a positive 6 and we see that we simplified
to x squared plus 5 x plus 6 so we can double check our factoring that way to make sure that we do it
right with our trinomial factored this way we can apply Zero Theorem and the zero theorem tells us that
when we have two factors multiplied together and they're equal to zero as long as one of the factors
is equal to zero.
This equation is going to be true and that should make sense to us because if we imagine that this X
plus three factor is equal to zero then we're going to get ZERO times quantity x plus two equal to zero.
In other words we're just saying X plus three equals zero.
So are substituting 0 for X plus 3.
If we do that we're going to get 0 times the quantity x plus 2.
We'll still just have zero on the left hand side here.
So we get zero equals zero and we can see that that's true.
We can also say that if we set X plus two equal to zero the other factor then we know that this equation
is going to be zero.
So what we would get is X plus three times zero equals zero.
And of course anything times zero is still 0 0 and get.
If you want to really make sure you can use the distributive property and distribute the zero across
as plus 3 and get 0 x plus 0 times 3 is zero.
So we get zero.
And as you can see we just have zero plus or equals zero or zero equals zero.
So what we're saying is that the zero of them allows us to set either factor equal to zero if either
factors equal to zero or if both factors are equal to zero.
We know this equation will be true.
We're going to set each of our factors separately equal to zero x plus three equals zero and X plus
two equals zero.
And then solve both of them per X and the way that we'll do that will subtract three from both sides.
We're going to say minus three on the left and minus three on the right and we're going to get positive
three negative three to cancel.
We're just left with X plus arrow because 0 is the identity number for addition.
We're just going to get X on the left hand side.
And over here on the right hand side we're just going to get negative 3.
So x is equal to negative 3 for our second one will do the same thing will subtract two from both sides
of our equation.
And what we'll get is positive to negative 2 to cancel on the left hand side x Passero is just X equal
to negative 2.
And now all we have are two solutions to our equation.
X equals negative 3 and X equals negative 2.
These are the solutions or the roots to the polynomial equation we were given.
And remember the reason that there are solutions is because if we plug in exit polls negative 3 here
for our first factor we'll get negative three plus three or zero.
So that will make this factor equal to zero.
We multiply zero times the quantity x plus to get zero on the left hand side equal to zero on the right
hand side and for here exit goes negative to that'll make our second factor equal to zero.
When we multiply zero times the X plus three factor will get 0 on the left.
So that's why those are the solutions to this equation.
If you want to check yourself even another way you can put both of these back into the original polynomial
equations.
For example if we wanted to plug negative three back into our original equation we should get zero on
the left hand side.
When we do that here we'll get negative 3 squared which is going to give us a positive 9 will get negative
3 times 5 which is going to give us a negative 15 and then we just have plus six equals zero.
We'll do our addition First we have nine plus six is 15.
So we get 15 minus 15 equals zero or zero equals zero.
And that checks out and we can do the same thing with negative to plug that back into our original polynomial
equation and we get zero.
So those are the solutions to our equation.
Let's do a second example quickly.
If we factor this will get X minus 2 times X minus 1 negative 2 in a negative one is going to give us
a negative 3 but negative 2 times and negative one gives us positive too which is why the factoring
works and you can double check that if you want to by multiplying it out.
So then we're going to set each of our factors equal to zero so we're going to say X minus two equals
zero and we're going to say X minus one equals zero adding two to both sides we get X equals a positive
to keep adding 1 to both sides we as equals positive one.
So we can say that the solutions or the roots to our equation are actually was two and X equals 1 for
this polynomial equation.