### Video Transcript

Find the dimensions of the box described. The length is three inches longer than the width, the width it two inches longer than the height, and the volume is 120 cubic inches.

To answer this problem, weโre going to need to form and solve an equation. Before we can form and solve an equation though, we need to define a variable and unknown. Weโre told that the length is longer than the width and the width is longer than the height. So letโs begin by defining the height. Let the height of the box be ๐ฅ inches. The width is two inches longer than the height, so this means itโs ๐ฅ plus two inches. Since the length is three inches more, then this is ๐ฅ plus two plus three inches, or ๐ฅ plus five inches. Thereโs one piece of information we havenโt yet used and thatโs that the volume of the box is 120 cubic inches.

We know that the volume of a cuboid is its length multiplied by its height multiplied by its width. So in this case, the volume is ๐ฅ times ๐ฅ plus two times ๐ฅ plus five. But thatโs equal to 120. So weโve got our equation. Letโs distribute our parentheses because in order to solve this, we need to ensure itโs equal to zero. Weโll begin by multiplying ๐ฅ plus two by ๐ฅ plus five. And when we do, we get ๐ฅ squared plus seven ๐ฅ plus 10. And thatโs because we began by multiplying the first two terms. ๐ฅ times ๐ฅ gives us ๐ฅ squared. We then multiply the outer term and the inner term. And we get ๐ฅ times five, which is five ๐ฅ, two times ๐ฅ, which is two ๐ฅ, and their sum is seven ๐ฅ.

Finally, we multiply the last terms. So we get two times five, which is 10. Weโll distribute again by multiplying each term by ๐ฅ. So we get ๐ฅ cubed plus seven ๐ฅ squared plus 10๐ฅ equals 120. Letโs subtract 120 from both sides. And we get ๐ฅ cubed plus seven ๐ฅ squared plus 10๐ฅ minus 120 equals zero. Now that we have a cubic expression which weโve made equal to zero, weโre going to need to factor that expression before we can solve. To factor, weโre going to need to find a factor of ๐ฅ cubed plus seven ๐ฅ square plus 10๐ฅ minus 120. So we recall the factor theorem. And this says that if ๐ of ๐ is equal to zero, then ๐ฅ minus ๐ is a factor of that function ๐ of ๐ฅ. So we need to find a value that when we substitute it into the expression ๐ฅ cubed plus seven ๐ฅ squared plus 10๐ฅ minus 120, we get zero.

We look for factors of negative 120. Those are numbers weโre going to try. And thatโs because our expression is going to factor to look like ๐ฅ plus ๐ times ๐ฅ plus ๐ times ๐ฅ plus ๐. And the constant term is the product of ๐ and ๐ and ๐. Now, in fact, ๐ฅ equals three is the factor of negative 120 weโre interested in. Three cubed plus seven times three squared plus 10 times three minus 120 is zero. So ๐ฅ minus three must be a factor. Weโre going to clear some space and perform polynomial long division to find the other factor or factors. There are a number of ways we can achieve this, but letโs use the bus stop method.

๐ฅ cubed divided by ๐ฅ is ๐ฅ squared. Then when we multiply squared by ๐ฅ minus three, we get ๐ฅ cubed minus three ๐ฅ squared. We now subtract ๐ฅ cubed plus three ๐ฅ squared from ๐ฅ cubed plus seven ๐ฅ squared. And we get 10๐ฅ squared. Letโs bring down the 10๐ฅ. We now divide 10๐ฅ squared by ๐ฅ to get 10๐ฅ. We multiply 10๐ฅ by ๐ฅ minus three to get 10๐ฅ squared minus 30๐ฅ. And then we subtract these two expressions, and thatโs 40 ๐ฅ. We bring down negative 120. And now we divide 40๐ฅ by ๐ฅ to get 40. We multiply 40 by ๐ฅ minus three, which is 40๐ฅ minus 120. And then we subtract. And that gives us zero. And thatโs what we would expect. We would expect no remainder because we saw the ๐ฅ minus three had to be a factor of our cubic.

So we can say that ๐ฅ minus three times ๐ฅ squared plus 10๐ฅ plus 40 is equal to zero. And in fact, we cannot factor ๐ฅ square plus 10๐ฅ plus 40. Itโs discriminate. Remember, for a quadratic equation of the form ๐๐ฅ squared plus ๐๐ฅ plus ๐ equals zero, the discriminate is ๐ squared minus four ๐๐. Its discriminate is 10 squared minus four times one times 40. Thatโs less than zero. So it means not only is it not factorable, but the equation ๐ฅ square plus 10๐ฅ plus 40 equals zero has no real roots. And so, for this equation to be equal to zero, ๐ฅ minus three itself has to be equal to zero, which means ๐ฅ is equal to three.

Remember, that was the height of the box. Its width was ๐ฅ plus two. So thatโs three plus two, which is five. And its length was ๐ฅ plus five. Thatโs three plus five, which is eight. So the dimensions of the box given are three, five, and eight inches. Now, in fact, we can perform a really quick check to see if what weโve done is correct. We find the volume of the box that weโve described, the one that has three, five, and eight inches as its dimensions. Thatโs three times five times eight, which is 120 cubic inches, as we expected. So we fully answered this question. Dimensions of the box are three, five, and eight inches.