How To Add Logarithmic Functions

divide x squared minus 3xplus 2 divided by x minus 2. so we're going todivide this into that. and we can do this reallythe same way that you first learned long division. so we have x minus2 being divided

How To Add Logarithmic Functions, into x squared minus 3x plus 2. another way wecould have written the same exact expression isx squared minus 3x plus 2, all of that over x minus 2.

that, that, and that areall equivalent expressions. now, to do this typeof long division-- we can call it algebraiclong division-- you want to look at the highestdegree term on the x minus 2 and the highest degree term onthe x squared minus 3x plus 2. and here's the x, andhere's the x squared. x goes into x squaredhow many times? or x squared dividedby x is what? well, that's just equal to x.

so x goes into xsquared x times. and i'm going to writeit in this column right here above all of the x terms. and then we want tomultiply x times x minus 2. that gives us-- xtimes x is x squared. x times negative2 is negative 2x. and just like you firstlearned in long division, you want to subtractthis from that. but that's completely thesame as adding the opposite,

or multiplying each ofthese terms by negative 1 and then adding. so let's multiplythat times negative 1. and negative 2x timesnegative 1 is positive 2x. and now let's add. x squared minus xsquared-- those cancel out. negative 3x plus 2x--that is negative x. and then we can bringdown this 2 over here. so it's negative x plus 2 leftover, when we only go x times.

so then we say, can x minus2 go into negative x plus 2? well, x goes into negativex negative one times. you can look at it right here. negative x dividedby x is negative 1. these guys cancel out. those guys cancel out. so negative 1 times x minus2-- you have negative 1 times x, which is negative x. negative 1 timesnegative 2 is positive 2.

and we want to subtractthis from that, just like you doin long division. but that's the same thingas adding the opposite, so negative x timesnegative 1 is positive x. positive 2 timesnegative 1 is negative 2. these guys cancelout, add up to 0. these guys add up to 0. we have no remainder. so we got this as beingequal to x minus 1.

and we can verify it. if we multiply x minus 1 timesx minus 2, we should get this. so let's actually do that. so let's multiply xminus 1 times x minus 2. so let's multiply negative2 times negative 1. that gives us positive 2. negative 2 times x--that's negative 2x. let's multiply xtimes negative 1. that is negative x.

and then x times x is x squared. and then add all the like terms. x squared, negative 2x minusx-- that's negative 3x. and then 2 plusnothing is just 2. and so we got thatpolynomial again.