Question Video: Factorizing the Difference between Two Cubes Mathematics

Factorize fully (𝑥 − 6𝑦)³ − 216𝑦³.

03:10

Video Transcript

Factorize fully 𝑥 minus six 𝑦 all cubed minus 216𝑦 cubed.

Whilst it might not be immediately obvious, this expression is written in the form 𝑎 cubed minus 𝑏 cubed. It is the difference of two cubes. We know that the factorization of 𝑎 cubed minus 𝑏 cubed is equal to 𝑎 minus 𝑏 multiplied by 𝑎 squared plus 𝑎𝑏 plus 𝑏 squared. The first term in our expression is 𝑥 minus six 𝑦 all cubed. This means that 𝑎 cubed is equal to this. As cube rooting is the opposite of cubing, we can cube root both sides of this equation, giving us a value of 𝑎 equal to 𝑥 minus six 𝑦. Our second term is 216𝑦 cubed; therefore, 𝑏 cubed is equal to this. Once again, we can cube root both sides of this equation, giving us 𝑏 is equal to six 𝑦 as the cube root of 216 is six.

We can now substitute our values of 𝑎 and 𝑏 into the right-hand side of the formula. We begin with 𝑎 minus 𝑏. This is equal to 𝑥 minus six 𝑦 minus six 𝑦. This simplifies to 𝑥 minus 12𝑦. 𝑎 squared will be equal to 𝑥 minus six 𝑦 all squared. This is equal to 𝑥 minus six 𝑦 multiplied by 𝑥 minus six 𝑦. Distributing our parentheses here gives us 𝑥 squared minus six 𝑦𝑥 minus six 𝑦𝑥 plus 36𝑦 squared. This can be simplified to 𝑥 squared minus 12𝑦𝑥 plus 36𝑦 squared. 𝑎𝑏 is equal to 𝑥 minus six 𝑦 multiplied by six 𝑦. Distributing the parentheses here gives us six 𝑦𝑥 minus 36𝑦 squared.

Finally, 𝑏 squared is equal to six 𝑦 all squared. This is equal to 36𝑦 squared. Substituting in the replacement for these three terms gives us 𝑥 squared minus 12𝑦𝑥 plus 36𝑦 squared plus six 𝑦𝑥 minus 36𝑦 squared plus 36𝑦 squared. This can be simplified to 𝑥 squared minus six 𝑦𝑥 plus 36𝑦 squared. Six 𝑦𝑥 is the same as six 𝑥𝑦. Therefore, the fully factorized form is 𝑥 minus 12𝑦 multiplied by 𝑥 squared minus six 𝑥𝑦 plus 36𝑦 squared.

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