Question Video: Expanding and Factorising Algebraic Expressions Involving Perfect-Square Trinomials Mathematics • 9th Grade

Expand π(π β 16π) + 64πΒ² β 81, and then factorise the result completely.

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Video Transcript

Expand π multiplied by π minus 16π plus 64π squared minus 81, and then factorise the result completely.

This question actually takes the form of two parts. The first part is where weβre gonna expand and in the second part where weβre gonna factor. So what weβre gonna do deal with is first part, which is expand. So what weβre gonna do is expand the parentheses and then actually simplify where we can before we move on to the second part. So the first term that weβre gonna have is π squared, and thatβs because we got π multiplied by π, which gives us π squared. And then weβre gonna minus 16 ππ, and we get this because π multiplied by negative 16π is negative 16ππ. And then we have our plus 64π squared and minus 81.

So now we take a look and can see well actually there arenβt any like terms. So we actually canβt simplify this any further, and weβve done the first part cause weβve actually expanded the expression. So now what we want to do is actually to move on and actually look at how weβd factor it. Well if weβre gonna look to actually factor the expression that weβve got, weβre actually going to start to use something called a perfect square trinomial, because actually three of our terms can help us to do that. Because if we actually look at the part which is π squared minus 16ππ plus 64π squared, then weβve got a relationship that we know when weβre looking at perfect square trinomials, because if weβve got something in the form π squared plus two multiplied by π multiplied by π plus π squared, then when you factor this, youβre gonna get π plus π multiplied by π plus π.

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. But how does this relate to the terms that I just mentioned in our expression? So first of all, we look at our relationship, and it says well something squared, so capital π΄ squared. Well in our case, what thatβs gonna be is actually just π squared. So that first term is nice and simple because weβve actually already got it in the form that is asked for.

But when we look at the final term, itβs going to be slightly different cause actually weβve got 64π squared. And what we say when we look at our relationship is that itβs again something squared, this time we called it capital π΅, but it means something else squared. 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. So we can say that itβs eight π and then all squared.

So now if we actually look at the middle term, we can see that we have negative and then weβve got two multiplied by π multiplied by eight π that we can actually see that weβre gonna get our negative 16ππ. Okay, great! So what we can actually see is weβve actually now got in the form of our second relationship. So therefore, in that case, it means that we now know how to factor. Because following the rule above, weβd actually have π minus π multiplied by π minus π. So in our case, thatβs our π minus eight π multiplied by π minus eight π.

Okay, great! So weβve actually factored the first three terms. So weβre now at the stage where we have π minus π multiplied by π minus eight π minus 81. But is this fully factored? Can we do anything else? Well if weβre actually dealing with squares, letβs have a look at the final part, which is negative 81. Well if you think about negative 81, this is the same as negative and then nine squared. So if we actually think about that, then that would be negative and then in parentheses nine squared, so negative 81.

So therefore, we think okay, great! Well letβs add this into our parentheses as part of the factoring. So Iβve done that, so weβve got π minus eight π, and Iβve got a nine in the first parentheses, and Iβve got π minus eight π and again nine in the second parentheses. However, we donβt want the nine to actually affect any of the terms. So how could we deal with this? Well the way that we actually deal with this is by having positive nine and negative nine, and thatβs because positive nine multiplied by negative nine is going to give us our negative 81 that we need to actually fulfill the original expression.

However, because we have positive nine and negative nine, any other multiplication that takes place are actually going to cancel themselves out. And Iβll show this by checking our answer using expansion. So first of all, we can have π multiplied by π, which gives us π squared. And then weβre going to have π multiplied by negative eight π, which gives us negative eight ππ. And then weβll have π multiplied by negative nine, which gives us negative nine π.

And then we move on to the second term in the first parentheses. So we have negative eight π multiplied by π, which gives us negative eight ππ. And then we have negative eight π multiplied by negative eight π, which gives is positive 64π squared. And then we have negative eight π multiplied by negative nine, which gives us positive 72π. So then we move on to the final term in the first parentheses. So we have positive nine multiplied by π, which gives us positive nine π. Then we have positive nine multiplied by negative eight π, which gives us negative 72π because we have a negative multiplied by a positive. And then finally, we have positive nine multiplied by negative nine, which gives us negative 81.

Okay, so weβve now fully expanded in our check. Now letβs try and actually simplify and collect like terms. Well thereβs only one π squared, so weβll have π squared. And then weβve got negative eight ππ and then minus eight ππ, which will give us negative 16ππ. Then we have negative nine π plus nine π, which is zero, so they cancel out. Then thereβs positive 64π squared. Then positive 72π minus 72π, which again will give us zero, so that cancels itself out. And then weβre left with negative 81.

So therefore, we get π squared minus 16ππ plus 64π squared minus 81. And a quick check back, and this is where we started. 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.