Factorize fully 𝑥 cubed plus eight 𝑦 cubed.
In this question, we are asked to fully factor a given algebraic expression. To do this, we need to start by looking at the expression we are asked to factor. We can see that the expression contains two terms and the variables in each term are raised to nonnegative integer exponents. Therefore, this is a binomial expression. This means that we need to fully factor a binomial.
We should always start by finding the greatest common divisor of all of the terms, since we can take out any factor shared by every term. In this case, we see that the first term has a coefficient of one. So we cannot take out any nontrivial shared constant factors. We can also see that the first term has no factor of 𝑦 and the second term has no factor of 𝑥. So we cannot take out any factors of 𝑥 or 𝑦. In other words, the greatest common divisor of the terms is one.
We are not done yet, since we can also check if this binomial resembles any special binomial that we already know how to factor. We can note that both the terms we are given are perfect cubes, since two 𝑦 all cubed is eight 𝑦 cubed.
We can then recall that we know how to factor the sum of two cubes. We know that 𝑎 cubed plus 𝑏 cubed is equal to 𝑎 plus 𝑏 times 𝑎 squared minus 𝑎𝑏 plus 𝑏 squared. Therefore, we can factor the given expression using the formula for factoring the sum of cubes by substituting 𝑎 equals 𝑥 and 𝑏 equals two 𝑦 into the formula. Doing this, we obtain 𝑥 plus two 𝑦 multiplied by 𝑥 squared minus 𝑥 times two 𝑦 plus two 𝑦 all squared.
We can evaluate the final term in the second factor by noting that two 𝑦 all squared is four 𝑦 squared to get the following expression. The three terms in the second factor have a greatest common divisor of one. And we cannot factor this expression further. So we obtain that 𝑥 cubed plus eight 𝑦 cubed is equal to 𝑥 plus two 𝑦 times 𝑥 squared minus two 𝑥𝑦 plus four 𝑦 squared.