### Video Transcript

factorize fully 𝑥 to the power of
five plus nine 𝑥 squared minus eight 𝑥 cubed minus 72.

In order to factorize this
expression, we need to put it into parentheses or brackets, by looking for common
factors. If we firstly look at the first two
terms, 𝑥 to the power of five plus nine 𝑥 squared, we need to try and find the
highest common factor. The highest common factor is 𝑥
squared. Therefore, we can take this out of
the parenthesis. 𝑥 squared multiplied by 𝑥 cubed
is 𝑥 to the power of five. And 𝑥 squared multiplied by nine
is nine 𝑥 squared. Therefore, we have 𝑥 cubed plus
nine inside the parenthesis.

We now need to find the highest
common factor of the last two terms, negative eight 𝑥 cubed minus 72. The highest common factor here is
negative eight. So we can take negative eight out
of the parenthesis. Negative eight multiplied by 𝑥
cubed is negative eight 𝑥 cubed. And negative eight multiplied by
positive nine is negative 72. At this stage, we notice that 𝑥
cubed plus nine is common to both parts of the expression. This means that our final answer is
𝑥 squared minus eight multiplied by 𝑥 cubed plus nine.

The full factorization of 𝑥 to the
power of five plus nine 𝑥 squared minus eight 𝑥 cubed minus 72 is 𝑥 squared minus
eight multiplied by 𝑥 cubed plus nine.

We could check this answer by using
the FOIL method to multiply out the parentheses. Multiplying the first terms, 𝑥
squared by 𝑥 cubed, gives us 𝑥 to the power of five. Multiplying the outside terms gives
us nine 𝑥 squared. Multiplying the inside terms gives
us negative eight 𝑥 cubed. And finally, multiplying the last
terms, negative eight and positive nine, gives us negative 72. As this gives us the expression
that we started with, we know that the factorization is correct.