In Exercises 1–16, divide using long division. State the quotient...

Question

In Exercises 1–16, divide using long division. State the quotient, and the remainder, r(x). (6x^3+7x^2+12x−5)÷(3x−1)

Accepted Answer

welcome back. I'm so glad you're here. We get to perform long division. So our dividend here is eight X cubed plus 22 X squared plus seven X plus five that goes inside the division bracket and it is divided by our divisor which is two X plus five. Alright, how do we do? Long division? We start off with step one which is to divide. Well that's not super helpful. Some what do we divide exactly? We're going to take the first term of our dividend divided by the first term of our divisor. So in this case X cubed divided by two X. An eight X cubed divided by two X is four X squared. I'm going to write that four X squared above the term in the dividend that has the same degree of X. So right above the other X squared. Ok, that's step one. Step two is to multiply. What do we multiply? We're going to take this first term in our quotient that we came up with and we're going to multiply it by every term in the divisor. So four X squared times two X. Which is going to give us eight x cubed and four X squared times five which will give us plus X squared. There is step two. Step 3 is to subtract we are going to subtract these two terms that we just came up with in our work from the first two terms in the dividend. So eight X cubed minus eight X cubed. Perfect. That's zero as it should be. I'm not going to write a zero and notice that I put both of these terms in parentheses so that we are sure to subtract both terms because now we go 22 X squared minus 20 X squared. And that will leave us with two X squared. All right. That's step three. Step four is to bring down well, what do we bring down? We're going to bring down the appropriate terms from our dividend so that we have the same number of terms in our work here. Two X squared plus seven X. That's two terms as in our divisor. The two X plus five. Because what we do next is start back over with step one. We are going to take the first term in our work. So two x squared and divide it by the first term in our divisor again. So divide that by two X. Two X squared divided by two X. Well, that is going to be just a plus one X. Now we continue with step two which is to multiply, we take our new term in our quotient and multiply it by both terms in the divisor X times two X is two X squared and X times five. That is plus five X. Now, resuming step three, subtract we're going to subtract both of these. We have two X squared minus two X squared. Perfect. That's zero. We did it correctly and seven X minus five X. Which will leave us with two X. Step forward to bring down and we're going to bring down that plus five and resuming with division, we're going to take the first term in our work here which is two X. And divided by the first term in the divisor two X. We have two X divided by two X, which is just one. So we'll write plus one in our quotient. Now continuing with the multiplication will take the one times both the two X and the 51 times two X is two X. One times five is plus five. Next step is to subtract, we're going to take two X minus two X. That's nothing great and five minus five. That is also nothing. That is zero. All right. When we can't continue with step four, When there's nothing left to bring down, then we have completed our long division and anything left here would be a remainder. In this case we have a remainder of zero, so there is no remainder. So we just have to look for the answer choice that matches our quotient. Our quotient is four X squared plus X plus one. Or in other words our answer is four X squared plus X plus one. We look at our answer choices and that matches with answer choice B. Well done. We'll catch you on the next one

In Exercises 1–16, divide using long division. State the quotient, and the remainder, r(x). (6x^3+7x^2+12x−5)÷(3x−1)

Key Concepts

Polynomial Long Division

Quotient and Remainder

Degree of a Polynomial