Out at you in the future- is you should always think about Immediately, hopefully what I'm about to say will make it pop Something that should hopefully pop out at you- if it doesn't We can use those factors as the denominators in each of theĬomponents, and a third-degree polynomial is much, much, much Going to decompose this into its components, we have toįigure out the factors of the denominator right here, so that But here, you can doįrom the highest degree term here isĪ second-degree term, here it's a third-degree term, Than the denominator, and if it's not, then you just do theĪlgebraic long division like we did in the first video. To just make sure that the numerator is of a lower degree Of these rational expressions that you want to decompose is Plus 20, all of that over x to the third minus 8. More complicated partial fraction decomposition problem. If anyone has any corrections or additions I'd love to hear them! So you could've come up with Sal's answer, which he got through long division, simply by plugging numbers into that formula (provided you could remember it! :) ). For the difference, we know that a^3 - b^3 = (a - b)*(a^2 + ab + b^2). In factoring we learnt a handy little shortcut formula for both the sums and differences of cubes. So, from the remainder theorem: p(2) = 0 x-2 is a factor of p(x).įurthermore, remember that x^3 - 8 is also a difference of cubes. You can pretty simply see that this value has to be 2. For our exercise, we know that, if we can figure out a value for x for which x^3 - 8 = 0, we'll have found our a and our corresponding factor x-a. The remainder theorem states that (x-a) is a factor of a polynomial p(x) if and only if p(a) = 0 (0 being the remainder here). Here's the relevant video: īasically, here's the rule: something is a factor of a polynomial p(x) if dividing the polynomial by that something gives you no remainder (this is true not just for polynomials but anything, really!). If you've been following the algebra 2 playlist you should've come across that. The way I understand it, it's best to see this as a consequence of the remainder theorem.
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