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Expand π multiplied by π minus 16π plus 64π squared minus 81, and then factorise the result completely.
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This question actually takes the form of two parts.
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The first part is where weβre gonna expand and in the second part where weβre gonna factor.
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So what weβre gonna do deal with is first part, which is expand.
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So what weβre gonna do is expand the parentheses and then actually simplify where we can before we move on to the second part.
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So the first term that weβre gonna have is π squared, and thatβs because we got π multiplied by π, which gives us π squared.
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And then weβre gonna minus 16 ππ, and we get this because π multiplied by negative 16π is negative 16ππ.
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And then we have our plus 64π squared and minus 81.
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So now we take a look and can see well actually there arenβt any like terms.
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So we actually canβt simplify this any further, and weβve done the first part cause weβve actually expanded the expression.
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So now what we want to do is actually to move on and actually look at how weβd factor it.
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Well if weβre gonna look to actually factor the expression that weβve got, weβre actually going to start to use something called a perfect square trinomial, because actually three of our terms can help us to do that.
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Because if we actually look at the part which is π squared minus 16ππ plus 64π squared, then weβve got a relationship that we know when weβre looking at perfect square trinomials, because if weβve got something in the form π squared plus two multiplied by π multiplied by π plus π squared, then when you factor this, youβre gonna get π plus π multiplied by π plus π.
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Or if you have π squared minus two multiplied by π multiplied by π plus π squared, this is gonna be equal to π minus π multiplied by π minus π.
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But how does this relate to the terms that I just mentioned in our expression?
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So first of all, we look at our relationship, and it says well something squared, so capital π΄ squared.
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Well in our case, what thatβs gonna be is actually just π squared.
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So that first term is nice and simple because weβve actually already got it in the form that is asked for.
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But when we look at the final term, itβs going to be slightly different cause actually weβve got 64π squared.
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And what we say when we look at our relationship is that itβs again something squared, this time we called it capital π΅, but it means something else squared.
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Well 64π squared can actually be written as something squared because that something can be eight π because eight multiplied by eight is 64 and π multiplied by π is π squared.
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So we can say that itβs eight π and then all squared.
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So now if we actually look at the middle term, we can see that we have negative and then weβve got two multiplied by π multiplied by eight π that we can actually see that weβre gonna get our negative 16ππ.
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Okay, great!
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So what we can actually see is weβve actually now got in the form of our second relationship.
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So therefore, in that case, it means that we now know how to factor.
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Because following the rule above, weβd actually have π minus π multiplied by π minus π.
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So in our case, thatβs our π minus eight π multiplied by π minus eight π.
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Okay, great!
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So weβve actually factored the first three terms.
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So weβre now at the stage where we have π minus π multiplied by π minus eight π minus 81.
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But is this fully factored?
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Can we do anything else?
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Well if weβre actually dealing with squares, letβs have a look at the final part, which is negative 81.
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Well if you think about negative 81, this is the same as negative and then nine squared.
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So if we actually think about that, then that would be negative and then in parentheses nine squared, so negative 81.
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So therefore, we think okay, great!
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Well letβs add this into our parentheses as part of the factoring.
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So Iβve done that, so weβve got π minus eight π, and Iβve got a nine in the first parentheses, and Iβve got π minus eight π and again nine in the second parentheses.
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However, we donβt want the nine to actually affect any of the terms.
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So how could we deal with this?
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Well the way that we actually deal with this is by having positive nine and negative nine, and thatβs because positive nine multiplied by negative nine is going to give us our negative 81 that we need to actually fulfill the original expression.
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However, because we have positive nine and negative nine, any other multiplication that takes place are actually going to cancel themselves out.
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And Iβll show this by checking our answer using expansion.
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So first of all, we can have π multiplied by π, which gives us π squared.
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And then weβre going to have π multiplied by negative eight π, which gives us negative eight ππ.
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And then weβll have π multiplied by negative nine, which gives us negative nine π.
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And then we move on to the second term in the first parentheses.
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So we have negative eight π multiplied by π, which gives us negative eight ππ.
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And then we have negative eight π multiplied by negative eight π, which gives is positive 64π squared.
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And then we have negative eight π multiplied by negative nine, which gives us positive 72π.
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So then we move on to the final term in the first parentheses.
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So we have positive nine multiplied by π, which gives us positive nine π.
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Then we have positive nine multiplied by negative eight π, which gives us negative 72π because we have a negative multiplied by a positive.
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And then finally, we have positive nine multiplied by negative nine, which gives us negative 81.
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Okay, so weβve now fully expanded in our check.
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Now letβs try and actually simplify and collect like terms.
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Well thereβs only one π squared, so weβll have π squared.
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And then weβve got negative eight ππ and then minus eight ππ, which will give us negative 16ππ.
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Then we have negative nine π plus nine π, which is zero, so they cancel out.
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Then thereβs positive 64π squared.
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Then positive 72π minus 72π, which again will give us zero, so that cancels itself out.
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And then weβre left with negative 81.
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So therefore, we get π squared minus 16ππ plus 64π squared minus 81.
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And a quick check back, and this is where we started.
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So therefore, we can say that if you expand π multiplied by π minus 16π plus 64π squared minus 81 and then factor the result completely, you get π minus eight π plus nine multiplied by π minus eight π minus nine.