Video Transcript
Complete the factorization. 1,000𝑥 cubed minus 729 is equal to 10𝑥 minus nine multiplied by what.
In this question, we are asked to complete the process of factoring an expression. To do this, we can start by looking at the expression that we want to factor. We can start by noting that the first term is a cube, since 1,000 is 10 cubed. We can therefore rewrite this term as 10𝑥 all cubed. Similarly, we can note that the second term is also a cube, since 729 is equal to nine cubed. This means that we are factoring a difference between cubes. Therefore, we can factor the expression by recalling the formula for factoring a difference of two cubes, which tells us that 𝑎 cubed minus 𝑏 cubed is equal to 𝑎 minus 𝑏 multiplied by 𝑎 squared plus 𝑎𝑏 plus 𝑏 squared.
To apply this result to factor the expression, we will set 𝑎 equal to 10𝑥 and 𝑏 equal to nine. Substituting 𝑎 equals 10𝑥 and 𝑏 equals nine into the formula gives us 10𝑥 minus nine multiplied by 10𝑥 squared plus 10𝑥 times nine plus nine squared.
We now want to simplify the expression inside the second set of parentheses. We have that 10𝑥 times 10𝑥 is 100𝑥 squared, 10𝑥 times nine is 90𝑥, and nine squared is 81. Therefore, we can continue factoring the expression to get 100𝑥 squared plus 90𝑥 plus 81.
It is worth noting that we can check our answer by distributing the factor of 10𝑥 minus nine over the parentheses to check that we get 100𝑥 cubed minus 729. Distributing the factor of 10𝑥 over the terms inside the second set of parentheses gives us 1,000𝑥 cubed plus 900𝑥 squared plus 810𝑥. We can then distribute the factor of negative nine over the second set of parentheses to obtain negative 900𝑥 squared minus 810𝑥 minus 729. Finally, we can add these expressions together to see that it is equal to 1,000𝑥 cubed minus 729. This is the same as the expression that we wanted to factor, so this confirms that the correct factor is 100𝑥 squared plus 90𝑥 plus 81.
