### Video Transcript

Factorize fully 1000 𝑥 cubed minus 125.

Our first step is to find the highest common factor of 1000 and 125. This is equal to 125 as 125 is the highest number that divides into 1000 and 125. Factorizing out 125 from our initial equation gives us 125 multiplied by eight 𝑥 cubed minus one.

The bit that is left in the bracket or parenthesis eight 𝑥 cubed minus one can be solved using the difference of two cubes. This states that 𝑎 cubed minus 𝑏 cubed is equal to 𝑎 minus 𝑏 multiplied by 𝑎 squared plus 𝑎𝑏 plus 𝑏 squared. In our example, 𝑎 is equal to two 𝑥 as two 𝑥 cubed is eight 𝑥 cubed and 𝑏 is equal to one as one cubed is equal to one.

Substituting in these values gives us 125 multiplied by two 𝑥 minus one multiplied by two 𝑥 all squared plus two 𝑥 multiplied by one plus one squared. Simplifying the second bracket gives us four 𝑥 squared plus two 𝑥 plus one. This means that our final answer is 125 multiplied by two 𝑥 minus one multiplied by four 𝑥 squared plus two 𝑥 plus one.

The full factorization of 1000𝑥 cubed minus 125 is 125 multiplied by two 𝑥 minus one multiplied by four 𝑥 squared plus two 𝑥 plus one.