The expression 𝑥 cubed plus 27 has two factors. One factor is 𝑥 plus three. What is the other factor?
𝑥 cubed plus 27 is something called a sum of cubes, where 𝑎 cubed plus 𝑏 cubed is equal to 𝑎 plus 𝑏 times 𝑎 squared minus 𝑎𝑏 plus 𝑏 squared. So our 𝑥 cubed plus 27, we know that one of the factors is 𝑥 plus three. And that matches with the 𝑎 plus 𝑏. So let’s go back to our original expression and make sure that 𝑎 should be equal to 𝑥 and 𝑏 should be equal to three.
So if 𝑎 cubed is equal to 𝑥 cubed and 𝑏 cubed is equal to 27, we can solve for 𝑎 and 𝑏 by taking the cube root of both sides of the equation, leaving 𝑎 to be equal to 𝑥, because the cube root of 𝑥 cubed is 𝑥. And now to solve for 𝑏, we take the cube root of both sides of the equation. And we find that 𝑏 is equal to three, because the cube root of 27 is three. And this is because 27 is equal to nine times three. And nine is equal to three times three. So three threes is equal to three cubed. So the cube root of three cubed is three.
So just as we said, 𝑎 is equal to 𝑥 and 𝑏 is equal to three. So to find the other factor, we simply need to plug 𝑎 and 𝑏 into this expression. 𝑎 squared will be equal to 𝑥 squared minus 𝑎 times 𝑏, so minus 𝑥 times three, plus 𝑏 squared.
So let’s go ahead and simplify this. And we get that it’s equal to 𝑥 squared minus three 𝑥 plus nine. This is what the other factor would be. So here’s what we started with. Here was one of the factors. And we solved for the missing factor. So once again, the missing factor is equal to 𝑥 squared minus three 𝑥 plus nine.