Video: Factorising by Grouping

Factorize fully π‘₯⁡ + 9π‘₯Β² βˆ’ 8π‘₯Β³ βˆ’ 72.

02:33

Video Transcript

factorize fully π‘₯ to the power of five plus nine π‘₯ squared minus eight π‘₯ 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, π‘₯ to the power of five plus nine π‘₯ squared, we need to try and find the highest common factor. The highest common factor is π‘₯ squared. Therefore, we can take this out of the parenthesis. π‘₯ squared multiplied by π‘₯ cubed is π‘₯ to the power of five. And π‘₯ squared multiplied by nine is nine π‘₯ squared. Therefore, we have π‘₯ cubed plus nine inside the parenthesis.

We now need to find the highest common factor of the last two terms, negative eight π‘₯ 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 π‘₯ cubed is negative eight π‘₯ cubed. And negative eight multiplied by positive nine is negative 72. At this stage, we notice that π‘₯ cubed plus nine is common to both parts of the expression. This means that our final answer is π‘₯ squared minus eight multiplied by π‘₯ cubed plus nine.

The full factorization of π‘₯ to the power of five plus nine π‘₯ squared minus eight π‘₯ cubed minus 72 is π‘₯ squared minus eight multiplied by π‘₯ cubed plus nine.

We could check this answer by using the FOIL method to multiply out the parentheses. Multiplying the first terms, π‘₯ squared by π‘₯ cubed, gives us π‘₯ to the power of five. Multiplying the outside terms gives us nine π‘₯ squared. Multiplying the inside terms gives us negative eight π‘₯ 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.

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