Divide using long division. State the quotient, and the remain...

Question

Divide using long division. State the quotient, and the remainder, r(x). $\frac{2x^5 - 8x^4 + 2x^3 + x^2}{2x^3 + 1}$

Accepted Answer

Hey everyone here we are asked to find the quotation of the expression using long division. So setting this up in long division formatting, we need to have the dividend underneath the bar. And for clarity I'm going to use placeholders for the missing powers. So we have four X to the fifth Power plus zero X to the fourth power plus two X cubed plus eight X squared plus zero X plus zero. And on the left hand side we need to include our divisor which is five X squared plus three. Next to start solving we take the leading term of our dividend and divided by the leading term of our divisor. So we have four X to the fifth power divided by five X squared. Now recalling the product rule for exponents as well as the rule for negative exponents, this expression simplifies to four x cubed divided by five. And now we take this term and add it to the top of the bar. So our first term of the quotient becomes four X cubed divided by five. And we also take this term and multiply it by our divisor. So we have four X cubed divided by five times the quantity of five X squared plus three. Now multiplying this out and simplifying we get four X to the fifth Power plus 12 X cubed divided by five. Now we have to subtract this resultant from the dividend. So we have minus the quantity of four X to the fifth power plus zero X to the fourth Power plus 12 X cubed divided by five. Now subtracting we have four minus +44 X to the fifth. So we get zero. Then we have zero minus zero for X to the fourth which is just zero. And finally to minus 12 5th for X to the third gives us minus two X to the third power divided by five. And now we just have to bring down the rest of the terms. So we also have plus eight X squared plus zero X plus zero. And now from here we repeat the process but we now take the initial term, the leading term of our remainder and again divided by the leading term of the divisor. So we have negative two X cubed divided by five divided by five X squared. In this expression simplifies to negative two X divided by 25. We then take this term and add it to the top of the bar giving us negative two X divided by 25. As the second term of the quotient. We then take the term and multiply it by the divisor. So we have negative two X divided by 25 times the quantity of five X squared plus three. Multiplying this out we get negative two X cubed divided by five minus six X divided by 25. We now subtract this resulted from the remainder. So we have minus of negative two X cubed divided by five plus zero X squared minus six X divided by 25. And now just subtracting we see that the x cubes cancel. So we have zero, we then have eight X squared minus zero X squared. So we get positive eight X squared and then we have zero x minus negative six X divided by 25. So we get positive six X divided by 25. Next just bringing down the last term which is just plus zero. We can now repeat the process taking the leading term which is eight X squared. And dividing it by the leading term of the divisor. So we have eight X squared divided by five X squared. And we can see that the x squares cancel. So we just have 8/5. We then add this 8/5 to the top of the bar. So our third term of the quotient is 8/5 and then we multiply the 8/5 by the divisor. So we have 8/5 times the quantity of five X squared plus three, multiplying this out. We get eight X squared plus 24 5th. Next we subtract this resultant from the remainder. So we have minus the quantity of a X squared plus zero X plus 24 5th. Now subtracting we see that we have eight x squared minus a X squared, giving us zero then six X divided by 25 minus zero X is just positive six X divided by 25. And lastly we have 0 -24/5 which gives us -24/5. And now we can see that the highest degree of our remainder is less than the highest degree of the divisor. So we are done dividing that tells us that this result here is our remainder and the top of the bar is the first three terms of our quotient. Therefore our quotient becomes four X cubed, divided by five minus two X, divided by 25 plus 8/5 plus the quantity of six X, divided by 25 minus 24 5th, all divided by the quantity of five X squared plus three. And this leaves us with answer choice. C. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

Divide using long division. State the quotient, and the remainder, r(x). $\frac{2x^5 - 8x^4 + 2x^3 + x^2}{2x^3 + 1}$

Key Concepts

Polynomial Long Division

Degree of a Polynomial

Quotient and Remainder in Polynomial Division

Divide using long division. State the quotient, and the remainder, r(x).2x5−8x4+2x3+x22x3+1\frac{2x^5 - 8x^4 + 2x^3 + x^2}{2x^3 + 1}2x3+12x5−8x4+2x3+x2

Key Concepts

Polynomial Long Division

Degree of a Polynomial

Quotient and Remainder in Polynomial Division

Divide using long division. State the quotient, and the remainder, r(x). $\frac{2x^5 - 8x^4 + 2x^3 + x^2}{2x^3 + 1}$