Where we have a term like this, we know that what the squared means is this bracket multiplied by itself. So now how do we multiply it out? We can use FOIL; so do the first term multiplied the first term, which will give us 𝑎 squared. Then 𝑎 multiplied by 𝑏, which will give us 𝑎𝑏. For the outer terms and the inner terms, which will be 𝑏 multiplied by 𝑎, which will give us 𝑎𝑏 again. And finally the last term 𝑏 multiplied by 𝑏 which gives us 𝑏 squared. So then if we collect the like terms, we can see that we’ve got two 𝑎𝑏. And in red underlined is called “the perfect trinomial.” So we’re saying for any 𝑎 and 𝑏 if we have 𝑎 plus 𝑏 all squared, that will give us 𝑎 squared plus two 𝑎𝑏 plus 𝑏 squared. And we’ll be able to factor expressions really easily using perfect trinomial as long as we can just spot what we’re doing. So let’s have a look at an example using perfect trinomial.
So let’s look at our first term first. Well we know that 𝑥 squared is 𝑥 all squared. And then looking at sixteen, we know that four squared is sixteen. So if our middle term eight 𝑥 follows the rule two multiplied by 𝑎 multiplied by 𝑏, where 𝑎 in this case is 𝑥 and 𝑏 is four, then we know that this is a perfect trinomial. And we can see it does. So we’ll be able to take 𝑎, which we can see is 𝑥, and 𝑏, which we can see is four, and then just put that straight in a bracket and we’ll square it. And there we have fully factored this expression.
so factor eighty-one 𝑥 to power four plus ninety 𝑥 squared plus twenty-five. Again we’re gonna look at our first term and our last term and try to work out if they are squares. So looking at the first term, we know that nine squared is eighty-one and we know that 𝑥 squared is 𝑥 to power four. So that means nine 𝑥 squared all squared is the same as eighty-one 𝑥 to power four. Right well let’s look at the last term. That’s easier; we can see that five squared is twenty-five. So then 𝑎 is nine 𝑥 squared and 𝑏 is five.
So if the middle term satisfies two multiplied by 𝑎 multiplied by 𝑏, then we know that it is a perfect trinomial. So let’s try it out. So two multiplied by five is ten; ten multiplied by nine is ninety. Well the coefficient works and then that’s 𝑥 squared. So-so does the variable. So this is a perfect trinomial. So we need to just pop them into the parentheses. So 𝑎 we can see is nine 𝑥 squared and 𝑏 is five. So we can see that our original expression is equal to nine 𝑥 squared plus five all squared. So all that — though that one looks a little bit tougher at the beginning, all we need to do is just have a look: is the first term squared? is the last term squared? And then do they satisfy the middle term as well? And that’s all you need to know for perfect trinomial.