In this video, we will look
at two examples of using the Shortcut Method
to factor a trinomial in the form x squared
plus b x plus c.
To factor a trinomial in this form,
we need to find two integers which we'll call r and s,
whose product is c, where c is the constant term
in the trinomial and whose sum is b,
where b is the coefficient of x
or the coefficient of the degree one term.
So if we can find the two integers
whose product is equal to c, and whose sum is equal to b,
then the factored form of x squared plus b x plus c
equals the quantity x plus r times the quantity x plus s.
And the order of these two binomial factors
does not matter, it can also be written as
the quantity x plus s times the quantity x plus r.
So looking at our first example, we have x squared
plus seven x minus 18, if this does factor,
it will factor into two binomial factors.
Next, notice c is equal to negative 18
and b is equal to positive seven.
So we're looking for two integers r and s,
whose product is equal to negative 18
and whose sum is equal to positive seven.
To help us determine the value of r and s,
let's list all the pairs of integers
that have a product of negative 18.
So we can have negative one times positive 18,
negative 18 times positive one,
negative two times positive nine,
negative nine times positive two,
as well as negative three times six,
or negative six times positive three.
So we're looking for the pair of integers
whose product is equal to negative 18
and has a sum of positive seven.
And therefore the integers we're looking for
are negative two and positive nine
because negative two plus nine is equal to positive seven.
Again these are the two integers that we want
because negative two times positive nine equals negative 18
and negative two plus nine is equal to positive seven.
So we'll let r equal negative two
and s equal positive nine, and this is what we need
in order to factor the given trinomial.
Because r is equal to negative two,
one factor will be trinomial would be the binomial
x plus negative two, or just x minus two
because s equals nine, another binomial factor
would be x plus nine.
This is the factored form of the given trinomial.
Before we look at the second example though,
let's multiply this out just to verify
this product equals the given trinomial.
We know in multiplying two binomials
we'll have four products.
One method is use repeated distribution.
To do this we distribute the first term
in the first binomial into the second binomial
which gives us two products,
and then we distribute the second term
in the first binomial into the second binomial
giving us two more products.
Distributing the x we have x times x which equals x squared.
Then we have x times nine, that's nine x, a plus nine x.
Then we distribute the negative two
and we have two times x equals negative two x,
or minus two x, and then negative two times positive nine
equals negative 18, or minus 18.
Notice how we have two like terms here.
Simplifying we get x squared plus seven x minus 18
which verifies this trinomial is factored correctly.
And now let's take a look at our second example.
We're given the trinomial y squared minus 11 y
plus 24, if this does factor,
it will factor into two binomial factors.
Next notice c is equal to positive 24
and b is equal to negative 11.
So we're looking for two integers
that have a product of positive 24
and have a sum of negative 11.
And again we'll call these two values r and s.
Notice how because the integers must have a
negative sum and a positive product,
both integers will need to be negative.
So let's list the pair of integers
that give us a product of positive 24
and again we know both integers must be negative.
So for example, negative one times negative 24,
negative two times negative 12,
negative three times negative eight,
or negative four times negative six.
Again we're looking for the integers
that have the sum of negative 11,
and therefore we're looking for the integers
negative three and negative eight.
Again because negative three times negative eight
equals positive 24, and negative three plus
negative eight equals negative 11.
So let's set r equal to negative three
and s equal negative eight.
So because r equals negative three one binomial factor
will be y plus negative three, or just y minus three,
because s equals negative eight, another binomial factor
will be y plus negative eight, or just y minus eight.
So this is the factored form of the given trinomial.
But again let's verify this by multiplying this out.
Again we have four products, one, two, three, and four.
Y times y equals y squared,
y times negative eight is negative eight y or minus eight y,
negative three times y equals negative three y
or minus three y, and finally
negative three times negative eight is equal to positive 24
so we have plus 24.
Combining like terms, negative eight y minus three y
equals negative 11 y, so we have y squared minus 11 y
plus 24, verifying this is the correct
factored form of the given trinomial.
I hope you found this helpful.