Worked example: finding the missing monomial factor

- [Voiceover] So we have negative 30x to the fifth is equal to negative 10x to the third, times F. And I encourage you to pause this video and see if you can figure out what F is going to be. Well, the way that we can tackle it, we could just isolate the F on the right-hand side here if we divide by negative 10x to the third. So we might say, well, we want to divide this side by negative 10x to the third, but if we want the equality to be true, if we want the left side to stay being the same as the right side, whatever we do with the right side we have to do to the left side as well. So we have to divide the left side by negative 10x to the third. And then what does that leave us with? Well, on the right-hand side, up top we're multiplying by negative 10x to the third, and then we're dividing by negative 10x to the third, well, multiplying by something and then dividing by that same thing is the same thing as just multiplying by one, or one way to think about it, they just cancel out. So we are just going to be left with an F. We're going to be left an F on the right-hand side. And there's the whole point, we wanted to solve for F. And on the left-hand side, we can first look at the coefficients. We could say negative 30 divided by, negative 30 divided by negative 10 is positive three. So that's going to be three. And then x to the fifth power divided by x to the third power, well that's going to be x-squared, x-squared. You could either think of it in terms of our exponent properties, we would subtract these two exponents, x to the five minus three, which is x-squared or you could say up on top that's x times x times x times x times x. Did I say that right? Five x's. You could do it as x, let me do to that, on top, you have x to the fifth which is this. I also like to remind myself why the exponent properties even work. And then on the denominator, on the denominator you have x times x times x and these three x's are going to cancel and you're just going to be left with x times x which is just x-squared. So you get F is equal to three x-squared so you could write, we could write that negative 30x to the fifth is equal to, is equal to the negative 10x to the third times F and now we know that F is three x-squared. Three x-squared. And so another way to describe what's going on in this equation, we could say that negative 30x to the fifth is divisible by either one of these factors. That negative 30x to the fifth is divisible by negative 10x to the third or we could say negative 30x to the fifth is divisible by three x-squared or we could say that three x-squared is a factor of negative 30x to the fifth. And the way that we can make these claims about factor and divisibility is we're dealing with non-fractional coefficients right over here and we're also dealing with non-fractional exponents right over here so that's why we're saying, hey, these are factors, this yellow thing and this magenta thing, factors of this blue thing or this blue thing is divisible by either one of these.