 When working with polynomials factorizations are going to be extremely important because we'll talk we're going to talk about this right now With the so-called factor theorem, but the X intercepts are going to correspond directly with the factors of the polynomial So the factor theorem tells us that if F is a polynomial function Then the factor X minus I should say the polynomial X minus C. It's a linear polynomial X minus C is a factor of f of X if and only if f of C equals 0 And so this tells us that there's this one-to-one correspondence with the X intercepts of the function So that's these things over here F of C equals zero means that C is an X intercept of the graph These X intercepts are one-to-one corresponds with the factors X minus C and so factoring polynomials is going to be critical for us as we try to Graph functions because we need to factor them to find the X intercepts This is essentially what we did for quadratic functions as well It's a consequence of the zero product property, but it turns out we can also go the other way around we can build polynomials by knowing their X intercepts are for polynomials the X intercepts are often called the roots of the Polynomials so if I say the root of the polynomial that means these are its X intercepts So can we find a polynomial of degree 3 whose roots are? Negative 3 2 and 5 well by the factor theorem if negative 3 is a root that means that X minus negative 3 is was a factor of the polynomial if 2 is a root that means that X minus 2 was a factor of the polynomial and a 5 is a root that means that X minus 5 is A factor of the polynomial and so this is our polynomial f of X now This isn't uniquely determined f of X because we don't have enough information to determine the y intercept which corresponds to this leading coefficient a We don't know what the leading coefficient is But we do know what these X intercepts are and so I would probably write this thing as a times X plus 3 Times X minus 2 times X minus 5 now if you want to we could multiply this polynomial out But we already have enough evidence to see that this polynomial is degree 3 because we have 1 2 3 factors Present right here, and so don't feel like ever Multiplying out of polynomials considered simplified Simplified is such an ambiguous term in mathematics because simplified really just means that the Expression the mathematical expression is in the simplest form in preparation for the next step if we don't know what the next step is simplified can be somewhat ambiguous and One thing we're gonna see is with polynomials factoring generally leads to simpler calculations So leaving something factored is actually going to be very favorable favorable for us And so we were able to construct a polynomial degree 3 given its three roots by the factor theorem because the roots of the polynomial correspond To the factors of the polynomial. There's just one to one correspondence there