### Video Transcript

Consider the shown rectangle. There are a couple of parts to this question. The first part is write an expanded expression for its area. And then, we will come on to the next part after this.

Well, in this question, what weโre trying to do is find an expression for the area of our rectangle. And we know that the area of a rectangle is equal to the length multiplied by the width. So therefore, what weโre gonna do is multiply the two dimensions weโve been given, which are a length of two ๐ฅ plus five and a width of ๐ minus two. So, if weโre gonna multiply these two values together, what weโre gonna do is distribute across our parentheses, or expand our brackets.

So, what this means is weโre multiplying one set of parentheses by the other. And to do that, what we do is we multiply each term in the left-hand parentheses by each term in the right-hand. So, weโre gonna start with two ๐ฅ multiplied by ๐, which is gonna give us two ๐๐ฅ. Then, weโve got two ๐ฅ multiplied by negative two, which is gonna give negative four ๐ฅ. So, thatโs the two ๐ฅ multiplied by both terms.

Now, we move on to the positive five. So, now, weโve got positive five ๐. And thatโs because positive five multiplied by ๐ is positive five ๐. And then, finally, what weโve got is minus 10. And thatโs because positive five multiplied by negative two is negative 10. Okay, great, so, now, what we do usually is simplify this. But we canโt simply this expression because we donโt have any like terms.

And thatโs because if we take a look, weโve got an ๐๐ฅ term, an ๐ฅ term, an ๐ term, and then weโve got a units term. Weโve got our negative 10. So therefore, none of them are like terms because none of them have the same power of ๐ or ๐ฅ. The only one that does, has both ๐ and ๐ฅ, so we canโt actually add this to any of the other terms. Right, so, now, weโve solved part A. And that is that the expanded expression for the area of our rectangle is two ๐๐ฅ minus four ๐ฅ plus five ๐ minus 10. Okay, great, now, letโs move on to the next part of the question.

So, now, for the second part of the problem, weโre told that the given rectangle is the base of a triangular prism of height three ๐ plus one. Write an expanded expression for the volume of the prism. Simplify the expression, if possible.

So, what Iโve done is drawn a quick sketch of our triangular prism. So, weโve got, weโve got the base, and weโve got the measurements that we know from the first part of question. And weโve got the height, which is three ๐ plus one. Well, in the question, we were asked to find the volume. And we know that the volume of a prism is equal to the area of the cross section multiplied by the length.

And in our prism, the cross section is a triangle. So, the first thing we want to do is work out the area of the triangle. And we know that the area of a triangle is equal to half the base times the height. In this case, half the base times the perpendicular height. So, the area of the triangle is gonna be equal to a half multiplied by ๐ minus two multiplied by three ๐ plus one.

So, the first thing I do is multiply each term in the first parentheses by a half. So, we get ๐ over two minus one multiplied by three ๐ plus one. Now, if we distribute across our parentheses, we get ๐ over two multiplied by three ๐, which gives us three ๐ squared over two. And then, we have ๐ over two multiplied by positive one, which just gives us ๐ over two. And then, negative one multiplied by three ๐ gives us negative three ๐. And then, finally, negative one multiplied by positive one gives us negative one.

So, great, weโve got three ๐ squared over two plus ๐ over two minus three ๐ minus one. And then, what we can do is we can collect the like terms. So, weโve got positive ๐ over two, so thatโs a half ๐, minus three ๐, which is gonna be minus two and a half ๐ or minus five over two ๐. So, now, weโve got that our area of our cross section because itโs three ๐ squared over two minus five over two ๐ minus one. So, now, what we can do is move on and find the volume of our prism.

So, to do that, what weโre gonna do is multiply the area of the cross section by the length. So, weโve got three ๐ squared over two minus five over two ๐ minus one multiplied by two ๐ฅ plus five. Even though weโve got three terms in our left set of parentheses, it doesnโt change how we distribute. We still multiply each term in the left by each term in the right.

So, weโre gonna start with three ๐ squared over two multiplied by two ๐ฅ, which is gonna give us three ๐ squared ๐ฅ. And thatโs because the twos will cancel cause weโve got two divided by two, which is just one. Then, weโve got three ๐ squared multiplied by ๐ฅ, which gives us three ๐ squared ๐ฅ. And then, this is gonna be plus 15 over two ๐ squared. And then, weโve got minus five ๐๐ฅ. And thatโs cause you got negative five over two ๐ multiplied by two ๐ฅ. Well, again, the twos would cancel out, so weโre left with negative five ๐๐ฅ. Then, weโve got minus 25 over two ๐.

And then, we move on to the negative one. Weโve got negative one multiplied by two ๐ฅ is negative two ๐ฅ. And then, finally, weโve got negative one multiplied by positive five, which gives us negative five. Well, now, if we check back to the question, it says simplify the expression, if possible. Well, can we simplify this? Well, the answer is no. And thatโs again because we donโt have any like terms.

Because weโve got an ๐ squared ๐ฅ term, an ๐ squared term, an ๐๐ฅ term, an ๐ term, an ๐ฅ term, and just a unit term. So, none of them are like terms. So therefore, we can say that the expression for the volume of the prism is three ๐ squared ๐ฅ plus 15 over two ๐ squared minus five ๐๐ฅ minus 25 over two ๐ minus two ๐ฅ minus five.