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