Zero product property

let's say that we've got the equation 2x minus 1 times X plus 4 is equal to 0 pause this video and see if you can figure out the x values that would satisfy this equation or essentially our solutions to this equation alright now let's work through this together so at first you might be tempted to multiply these things out or there's multiple ways that you might have tried to approach it but the key realization here is that you have two things being multiplied and it's being equal to 0 so you have the first thing being multiplied is 2x minus 1 this expression is being multiplied by X plus 4 and to get it to be equal to 0 1 or both of these expressions needs to be equal to 0 let me really reinforce that idea if I had two variables let's say a and B and I told you a times B is equal to 0 well can you get the product of two numbers to equal 0 without at least one of them being equal to 0 and the simple answer is no if a is 7 the only way that you would get 0 is if B is 0 or if B was 5 the only way to get zeros if a is 0 so you see from this example either let me write this down either A or B or both because 0 times 0 is 0 or both must be 0 the only way that you get the product of two quantities and you get 0 is if one or both of them is equal to 0 I really want to reinforce this idea I'm going to put a I'm gonna put a red box around it so that it really get stuck in your brain I want you to think about why that is try to come up with two numbers try to multiply them so that you get zero and you're going to see that one of those numbers is going to need to be is going to need to be 0 so we're going to use this idea right over here now this might look a little bit but you could view 2x minus 1 as our a and you could view X plus 4 as our B so either 2x minus 1 needs to be equal to 0 or X plus 4 needs to be equal to 0 or both of them needs to be equal to 0 so I could write that as 2 X minus 1 needs to be equal to 0 or or X plus 4 or X let me do that Orange actually let me do the 2 X minus 1 in that yellow color so either 2x minus 1 is equal to 0 or X plus 4 is equal to 0 X plus 4 is equal to 0 and so let's solve each of these if 2x minus 1 could be equal to 0 well see you could add 1 to both sides and we get 2x is equal to 1 divide both sides by 2 and this is just straightforward solving a linear equation if this looks unfamiliar I encourage you to watch videos on solving linear equations on Khan Academy but you'll get X is equal to 1/2 as one solution this is interesting because we're going to have two solutions here or over here if we want to solve for X we can subtract 4 from both sides and we would get X is equal to negative 4 so it's a need in an equation like this you can actually have two solutions X could be equal to 1/2 or X could be equal to negative 4 I think it's pretty interesting to into substitute either one of these in if X is equal to 1/2 what is going to happen well this is going to be 2 times 1/2 minus 1/2 times 1/2 minus 1 that's going to be our first expression and that our second expression is going to be 1/2 plus 4 and so what's this going to be equal to well 2 times 1/2 is 1 1 minus 1 is 0 so I don't care what you have over here 0 times anything is going to be equal to 0 so when what x equals 1/2 the first thing becomes 0 making everything making the product equal zero and likewise if x equals negative four it's pretty clear that this second expression is going to be zero and even though this first expression isn't going to be zero in that case anything times zero is going to be zero let's do one more example here so let me delete out everything that I just wrote here and so let's I'm gonna involve a function so let's say someone told you that f of X is equal to X minus five times 5x plus two and someone said find the zeros of f of X well the zeros are what are the X values that make f of X equal to zero when does f of X equal zero for what for what X values does f of X equal zero that's what people are really asking when they say find the zeros of f of X so to do that well when does f of X equal zero well f of X is equal to zero when this expression right over here is equal to zero and so it sets up just like the equation we just saw X minus five times 5x plus 2 when does that equal zero and like we said like we saw before this says well this is just like what we saw before and I encourage you to pause the video and try to work it out on your own so there's two situations where this could happen where either the first the first expression equals zero or the second expression or maybe in some cases you'll have a situation where both expressions equal zero so we could say either X minus five is equal to zero or 5x plus two is equal to zero I'll write in or right over here now if we solve for X you add five to both sides of this equation you get X is equal to five here let's see to solve for X you can subtract two from both sides you get 5x is equal to negative two you can divide both sides by five to solve for x and you get X is equal to negative two-fifths so here are our two zeros you input either one of these into f of X if you input X equals five if you take F of five if you try to evaluate F of five then this first expression is going to be zero and so a product of zero and something else it doesn't matter that this is going to be twenty seven zero times 27 is zero and if you take F of negative two-fifths it doesn't matter what this first expression is the second expression right over here is going to be zero zero times anything is zero