In Exercises 27–29, divide using long division. (4x^3 - 3x^2 - 2x...

Question

In Exercises 27–29, divide using long division. (4x^3 - 3x^2 - 2x + 1) ÷ (x + 1)

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Hello. Today we're going to be simplifying a polynomial using long division. So the polynomial given to us is five X to the fourth? Power minus 13 X cubed minus 13 X squared plus 22 X minus two. And all of that is being divided by x minus three. So how do we start this process? Well a trick is to go ahead and multiply this mono meal by a term that will allow X to be as close as possible to the first term of the polynomial. So what can we multiply X by to give us a number as close as possible to five X to the fourth. Well X times five X cubed is going to give us five X to the fourth. So again if we multiply this entire mono meal By five x cubed What we are left with is five X to the 4th -15 x cubed. Now keep in mind the rules of division tell us that we are subtracting this value so in order to subtract this entire value we need to go ahead and distribute this negative sign into this mono meal. Doing so is just going to change the sign of each term. And what we are left with is negative five X to the fourth plus 15 X cubed. And now what we need to do is go ahead and simplify while five X to the fourth minus five X to the fourth, cancel each other out and negative 13 X cubed plus 15 X cubed gives us two X cubed. Now we're gonna go ahead and bring down our next term which in this case is negative 13 X squared. And we're gonna go ahead and repeat this process. So what's a number that I can multiply X by to get me as close as possible to to execute? Well X times two X squared gives me two X cubed. So if I multiply this mono meal by positive two X squared. What that leaves me with is two X cubed minus six X squared. And just like before we are subtracting this value so we're gonna have to distribute that negative sign. Doing so just changes the terms and what we are left with is negative two X cubed plus six X squared and just go ahead and simplify two X cubed minus two X cubed cancels each other out negative 13 X squared plus six X squared. Leaves me with negative seven X squared. Bring down the next term And that's plus 22 x. What's the number that I can multiply X by to get me as close as possible To negative seven X squared. Well X times negative seven X gives me negative seven X squared. So if I multiply this mono meal by negative seven X. What I'm left with is negative seven X squared plus 21 X. And once again we're going to have to distribute a negative so doing so it's just going to change the signs of the terms. And what we are left with is positive seven X squared minus 21 X negative seven X squared plus seven X squared cancel each other out and 22 x minus X leaves me with positive X. I'm gonna go ahead and bring down the next term which is minus two. And finally what's the number that I can multiply X by to give me X. Well that number is just going to be positive one because X times one gives me X. So what we are left with now is X -3 and once again we are subtracting this value so we need to go ahead and change the terms and what that gives me is negative X plus three. So X minus x is going to cancel each other out and negative two plus three gives me one. Now one in this case is going to be a remainder. So we're gonna need to find a way to include that into our solution. So let's go ahead and just list out our solution now our solution is going to be our quotient plus our remainder. So writing everything out what we got after doing the long division is five x cubed plus two X squared minus seven X plus one and now we need to include the remainder. Keep in mind that this value is positive so we're adding the remainder to our quotient and the remainder is still being divided by our original mono meal. So what we're adding to our quotient is going to be a remainder, which is this case one divided by x minus three, and this is going to be our solution, which also means that the solution to this problem is going to be a so I hope this video helps you in understanding how to perform long division, and I'll go ahead and see you all in the next video.

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