Perform each division. See Examples 9 and 10. (10x^3+11x^2-2x+3)/(5x+3)

Question

Accepted Answer

Hello, today, we're going to be simplifying the given quantity by using long division. So what we are given is 20 B cubed plus 22 B squared minus four B plus three. And we're gonna be dividing this quantity by two B plus three. Now, in their current numerator, we are not missing any terms as we have A B cubed term B squared B to the first and constant term. So we can go ahead and set up the long division immediately. We're going to be dividing 20 B cubed plus 22 B squared minus four B plus three. And once again, we're dividing this quantity by two B plus three. Now, in order to start this process, you want to go ahead and ask yourself what is a number that I can multiply two B plus three to get the first term to be as close as possible to 20 B cubed. Well, what if we multiply this quantity by 10 B squared? If we multiply two B plus three by 10 B squared, we're gonna go ahead and distribute this quantity into two B plus three. And this is going to give us 20 B cubed plus B squared. And that matches up with the first term as this product gives us 20 B cubed. And the first term in the division is 20 B cubed. So let's go ahead and put this underneath. So again, we have 20 B cubed plus 30 B squared. And we want to go ahead and be careful with our setup because we're trying to subtract this quantity. And since we're subtracting this quantity, we're going to need to distribute the negative sign into 20 B cubed plus 30 B squared. Doing this is just gonna go ahead and change the signs. And what we now have is negative 20 B cubed minus 30 B squared. And we're gonna go ahead and start the simplification, 20 B cubed minus 20 B cubed is just going to cancel itself out. And 22 B squared minus 30 B squared is going to give us negative eight B squared. We're gonna go ahead and bring down the next term which is negative four B and we're gonna go ahead and repeat this process now again, what's a number that I can multiply two B plus three by to get to be as close as possible to negative eight B squared. While we can multiply this quantity by negative four B if we multiply two B plus three by negative four B. Once again, we're distributing negative four B into two B plus three. And that's going to give us the quantity, negative eight B squared minus 12 B. We're gonna go ahead and put this underneath negative eight B squared minus 12 B. And again, be careful with your setup because you want to subtract this quantity. So you're going to need to distribute the negative sign into negative eight B square minus B. And doing this is just going to change the signs. So what we now have is positive eight B squared plus 12 B and doing the simplification while negative eight B squared plus 80 square is gonna cancel each other out and negative four B plus 12 B is going to give me positive eight B. We're gonna bring down the next term which is positive three and we're going to repeat this process. And again, what's a number that can multiply two B plus three to get to be as close as possible to eight B while I can multiply this quantity by positive four as multiplying two B plus three by positive four is going to give us eight B plus 12. So we're gonna go ahead and put that underneath and we're going to want to subtract that quantity which again is going to require us to distribute the negative sign into eight B plus 12 and distributing a negative sign is going to change the signs and we have negative eight B minus 12, eight B minus eight B is going to cancel each other out and negative 12 plus three. Is going to give me negative nine and this is going to be our remainder. So let's go ahead and write the answer. We start writing the answer by first writing the question we came up with and the quotient is going to be 10 B squared minus four B plus four. And we need to include the remainder into this quantity. Well, the remainder is a negative number. So we're going to be subtracting the remainder from this entire quantity here and we're going to write the remainder in the form of a fraction are actual remainder goes on the numerator which is going to be negative nine, but I move the negative in front of this quantity and we're gonna keep the same denominator as the original problem. And the original denominator was to be plus three and this is going to be the quotient or the simplified form of the original quantity through long division. And with that being said, the answer to this problem is going to be c so I hope this video helps you in understanding how to do long division. And I'll go ahead and see you all in the next video.

Perform each division. See Examples 9 and 10. (10x^3+11x^2-2x+3)/(5x+3)

Verified Solution

Key Concepts

Polynomial Division

Synthetic Division

Remainder Theorem