Video Transcript
Complete the following: Blank is
equal to 𝑦 plus 15𝑥 multiplied by 𝑦 squared minus 15𝑦𝑥 plus 225𝑥 squared.
Our first thought in this question
might be to try and distribute the parentheses, to multiply 𝑦 squared minus 15𝑦𝑥
plus 225𝑥 squared firstly by 𝑦 and then by 15𝑥. However, we might notice that our
expression is written in the form 𝑎 plus 𝑏 multiplied by 𝑎 squared minus 𝑎𝑏
plus 𝑏 squared. This is the factored form of the
expression 𝑎 cubed plus 𝑏 cubed. This is known as the sum of two
cubes. Our value of 𝑎 is 𝑦, and our
value of 𝑏 is 15𝑥.
We can work out the value of 𝑎
cubed and 𝑏 cubed by cubing both sides of each of these equations. Our first equation gives us 𝑎
cubed is equal to 𝑦 cubed. Cubing both sides of our second
equation gives us 𝑏 cubed is equal to 3,375𝑥 cubed. This is because 15 multiplied by 15
multiplied by 15 is 3,375. The missing term is, therefore, 𝑦
cubed plus 3,375𝑥 cubed as this is equal to 𝑦 plus 15𝑥 multiplied by 𝑦 squared
minus 15𝑦𝑥 plus 225𝑥 squared.
We would’ve got the same answer had
we distributed the two sets of parentheses. All the terms would’ve canceled
with the exception of 𝑦 multiplied by 𝑦 squared, which is 𝑦 cubed, and 15𝑥
multiplied by 225𝑥 squared, which is 3,375𝑥 cubed.
