Question Video: Multiplying Linear Expressions Mathematics

Video Transcript

Consider the shown trapezoid. Write an expanded expression for its area. Simplify the expression, if possible.

There’s also a second part of the question that we’ll come on to. So what we’re looking at here in this question is the area of our trapezoid. So we need to know a formula for this. Well, we know the formula for the area of a trapezoid. And that is the area is equal to a half 𝑎 plus 𝑏 multiplied by ℎ. And this is where 𝑎 and 𝑏 are the parallel sides of our trapezoid and ℎ is the distance between them.

So in our trapezoid, we got 𝑎, 𝑏, and ℎ, which I’ve labeled. It doesn’t matter which one, the top or the bottom, is 𝑎 or 𝑏. These are interchangeable. So when we put it together, we’re gonna get 𝑎 is equal to a half multiplied by 𝑎 minus five plus 𝑎 plus three multiplied by 𝑎 minus four, which is gonna give us a half multiplied by two 𝑎 minus two multiplied by 𝑎 minus four. So what we could do here is we could distribute across the parentheses and then half the result. But what I’m gonna start with is by halfing the first parentheses. And then we’re gonna distribute across our parentheses.

So when I do that, I’m gonna get 𝑎 minus one multiplied by 𝑎 minus four. Well, if we check what the question wants, the question wants us to expand the expression and then simplify it. So first of all, we’re gonna expand or we’re going to distribute across our parentheses. And when we do this, what we’re gonna get is 𝑎 multiplied by 𝑎, which is 𝑎 squared. 𝑎 multiplied by negative four is negative four 𝑎. Negative one multiplied by 𝑎 is negative 𝑎. And negative one multiplied by negative four, which is positive four.

So now, if we collect our like terms, what we’re gonna be left with is 𝑎 squared minus five 𝑎 plus four since this is our expanded expression for the area of our trapezoid. So now what we’re gonna do is we’re gonna move on to the second part of the question.

So for the second part of the question, we’re told that the given trapezoid is a cross section of the given prism of length two 𝑎 minus five. What we need to do is write an expanded expression for the volume of the prism, then simplify the expression, if possible.

Well, we have the formula for the prism. And that formula is that the volume is equal to the cross-sectional area or the area of the cross section multiplied by the length. Well, we know that the area of the cross section is equal to 𝑎 squared minus five 𝑎 plus four cause we’ve found that in the first part of the question. And we’re told in the second part of the question that the length is equal to two 𝑎 minus five. So therefore, we know the volume is gonna be equal to 𝑎 squared minus five 𝑎 plus four multiplied by two 𝑎 minus five.

Now if we distribute across our parentheses, first of all, we’re gonna have 𝑎 squared multiplied by two 𝑎, which is two 𝑎 cubed, and 𝑎 squared multiplied by negative five, which is negative five 𝑎 squared. So then we’re gonna get negative 10𝑎 squared plus 25𝑎. Then finally, we’re gonna get positive eight 𝑎 minus 20.

Okay, now what we need to do is simplify the expression. And we do that by collecting our like terms. So then when we’ve done that, we get that the volume is equal to two 𝑎 cubed minus 15𝑎 squared plus 33𝑎 minus 20. So there we’ve found the area of the cross section and the volume, and we’ve simplified them to these two expressions.