### Video Transcript

In this video, we will learn how to
factor quadratic expressions in the form ππ₯ squared plus ππ₯ plus π, where the
coefficient of the leading term is one. We will begin by defining what we
mean by a quadratic expression.

A quadratic trinomial is an
algebraic expression in the form ππ₯ squared plus ππ₯ plus π. π is the leading or quadratic
coefficient, π is the linear coefficient, and π is the constant term. A monic quadratic trinomial, on the
other hand, is an expression in the form π₯ squared plus ππ₯ plus π. The leading or quadratic
coefficient is equal to one. In this video, we will look at
quadratics of this type that can be factored into two sets of parentheses or
brackets. We will rewrite π₯ squared plus
ππ₯ plus π in the form π₯ plus π multiplied by π₯ plus π.

If we recall our FOIL method for
distributing two sets of parentheses, multiplying the first terms gives us π₯
squared. Multiplying the outside or outer
terms gives us ππ₯. Multiplying the inner terms gives
us ππ₯. Finally, multiplying the last terms
gives us ππ. We can simplify the middle two
terms by factoring out an π₯. ππ₯ plus ππ₯ is equal to π plus
π multiplied by π₯. This means that the expansion of π₯
plus π multiplied by π₯ plus π is equal to π₯ squared plus π plus π multiplied
by π₯ plus ππ.

When comparing this with π₯ squared
plus ππ₯ plus π, the π has been replaced by π plus π. This means that the linear
coefficient π must be equal to π plus π. In the same way, the constant term
π must be equal to π multiplied by π. We can therefore conclude that when
factoring a monic quadratic, the sum of the numbers π and π must equal π and the
product of π and π must equal π.

We will now look at some questions
where we need to factor a monic quadratic.

Factor π₯ squared plus eight π₯
plus 12.

In this question, we have a
quadratic of the form π₯ squared plus ππ₯ plus π. As the leading coefficient or
quadratic coefficient is equal to one, this can be factored into two sets of
parentheses, π₯ plus π multiplied by π₯ plus π. For any quadratic of this type, π
is equal to π plus π and π is equal to π multiplied by π. We are looking for two numbers that
have a product of 12 and a sum of eight. The number 12 has three sets of
factor pairs: one and 12, two and six, and three and four. Of these three pairs, only two and
six also sum to eight. The expression π₯ squared plus
eight π₯ plus 12 in its factored form is equal to π₯ plus two multiplied by π₯ plus
six.

We can check this answer by
expanding the brackets or distributing the parentheses. One way of doing this is using the
FOIL method. Multiplying the first terms gives
us π₯ squared. Multiplying the outer terms gives
us six π₯. Multiplying the inner terms gives
us two π₯. And finally, multiplying the last
terms gives us 12. Collecting like terms, this
simplifies to π₯ squared plus eight π₯ plus 12. As this is the same as our original
expression, we know that our answer π₯ plus two multiplied by π₯ plus six is
correct.

In this question, both π and π
were positive. In our next example, we will look
at a case where π and π are negative.

Factor π₯ squared minus eight π₯
minus 20.

This quadratic expression has a
leading coefficient of one. This means it is of the type π₯
squared plus ππ₯ plus π. If this quadratic can be factored,
we can write it as the product of two sets of parentheses, π₯ plus π multiplied by
π₯ plus π. The sum of π plus π is equal to
the coefficient of π₯, π. The product of π and π is equal
to the constant term π.

We need to find two numbers that
have a product of negative 20 and a sum of negative eight. Letβs begin by considering the
factor pairs of 20. They are one and 20, two and 10,
and four and five. As we want the product of these
numbers to be negative, one must be positive and one must be negative. There is no way that the pairs one
and 20 and four and five can lead us to a sum of negative eight. If we had negative two and positive
10, these would have a sum of positive eight. This means that we actually want
positive two and negative 10. These two numbers have a product of
negative 20 and a sum of negative eight. This means that the factored form
of π₯ squared minus eight π₯ minus 20 is π₯ plus two multiplied by π₯ minus 10.

We could check this answer by
distributing our parentheses. Multiplying π₯ by π₯ gives us π₯
squared. Multiplying π₯ by negative 10 gives
us negative 10π₯. Two multiplied by π₯ is equal to
two π₯. And two multiplied by negative 10
is negative 20. As this simplifies to π₯ squared
minus eight π₯ minus 20, which is the same as our original expression, we know that
our answer is correct.

We will now look at a more
complicated problem where we need to expand or distribute our parentheses first.

Expand and simplify π multiplied
by π plus nine plus 18, then factorize the result.

In order to expand the brackets or
distribute the parentheses, we need to multiply π by π and π by nine. This gives us π squared plus nine
π. We can drop down the constant term
18 so that π multiplied by π plus nine plus 18 is equal to π squared plus nine π
plus 18. This is a quadratic expression
written in the form π₯ squared plus ππ₯ plus π. The leading coefficient of the
quadratic is one. This means that it can be factored
into two sets of parentheses, π₯ plus π multiplied by π₯ plus π.

The first term in both of our
parentheses will be π as π multiplied by π is π squared. The numbers π and π must sum to
give us π, and they must multiply or have a product of π. We need to find two numbers that
have a sum of nine and a product of 18. There are three pairs of integers
that multiply to give us 18: one and 18, two and nine, and three and six. The only one of these pairs that
has a sum of nine is three and six. π squared plus nine π plus 18 is
equal to π plus three multiplied by π plus six. This means that our initial
expression π multiplied by π plus nine plus 18 is also equal to π plus three
multiplied by π plus six.

Our next example is a bit of a
trick question as we actually start with a cubic equation.

Factorize fully π₯ cubed plus two
π₯ squared minus 63π₯.

This expression is a cubic as the
highest exponent is a three. This suggests that it could be
quite difficult to factor or factorize. However, when we look at the
expression more closely, we notice that all three terms have a common factor of
π₯. This means that we can take this
factor outside our parentheses. π₯ cubed divided by π₯ is equal to
π₯ squared. Two π₯ squared divided by π₯ is
equal to two π₯. Finally, negative 63π₯ divided by
π₯ is negative 63.

We now have a quadratic expression
of the form π₯ squared plus ππ₯ plus π. As the coefficient of π₯ squared is
equal to one, we can try and factor this into two sets of parentheses, π₯ plus π
multiplied by π₯ plus π. The sum of π and π will be equal
to π, and the product of π and π will be equal to π. We need to try and find two
integers with a product of negative 63 and a sum of two. We know that nine multiplied by
seven is equal to 63. Multiplying a positive number by a
negative number gives a negative answer. Therefore, positive nine multiplied
by negative seven or negative nine multiplied by positive seven both give us
negative 63. As we want our numbers to have a
sum of positive two, the correct pair is positive nine and negative seven.

The expression π₯ squared plus two
π₯ minus 63 is equal to π₯ plus nine multiplied by π₯ minus seven. This means that the fully
factorized form of π₯ cubed plus two π₯ squared minus 63π₯ is equal to π₯ multiplied
by π₯ plus nine multiplied by π₯ minus seven. We could expand this expression by
distributing our parentheses to check that our answer is correct.

In our next question, we need to
work out the missing term in a quadratic expression.

Given that the expression π₯
squared plus ππ₯ minus 35 can be factorized, what is the set of all possible
integer values of π?

The expression in this question is
written in the form π₯ squared plus ππ₯ plus π. The leading coefficient or
coefficient of π₯ squared is equal to one. We are told that it can be factored
or factorized. This means we can write it in the
form π₯ plus π multiplied by π₯ plus π. The coefficient of π₯, the value
π, will be equal to π plus π. And the constant term π will be
equal to π multiplied by π. This means that in our expression,
the two values π and π must have a sum of π and a product of negative 35. The number 35 has two factor pairs:
one and 35 and five and seven.

As multiplying a negative number by
a positive number gives a negative answer, there are four pairs of integers that
multiply to give us negative 35. We have negative one and positive
35, positive one and negative 35, negative five and positive seven, and five and
negative seven. These are the potential values of
π and π. We need to find the sum of each of
these pairs of values as we know that the coefficient of π₯ will be equal to π plus
π. Negative one plus 35 is equal to
34. The sum of one and negative 35 is
negative 34. Negative five plus seven is equal
to two. And five plus negative seven is
equal to negative two. This means that the set of all
possible integer values of π in ascending order are negative 34, negative two, two,
and 34.

These values occur when our sets of
parentheses are π₯ minus 35 multiplied by π₯ plus one, π₯ minus seven multiplied by
π₯ plus five, π₯ minus five multiplied by π₯ plus seven, and π₯ minus one multiplied
by π₯ plus 35, respectively. After expanding these brackets, the
π₯-term would become negative 34π₯, negative two π₯, two π₯, and 34π₯, giving our
values of π of negative 34, negative two, two, and 34.

We will now summarize the key
points from this video. A monic quadratic expression has a
leading coefficient of one. This means it can be written in the
form π₯ squared plus ππ₯ plus π. If an expression of this form can
be factored, it can be written π₯ plus π multiplied by π₯ plus π. The coefficient of π₯ is equal to
π plus π, and the constant term is equal to π multiplied by π. This means that in order to factor
an expression of this type, we need to find two integers that have a product equal
to π and a sum equal to π. Using this information, we also saw
how we could find the missing term in a quadratic expression.