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). (x^2+8x+15)÷(x+5)

Accepted Answer

welcome back. I'm so glad you're here. We get to perform long division. We start off with our dividend which is X squared plus 12 X plus 35. Put that inside the division bracket. That is all divided by our divisor which is X plus seven. Alright, so how do we perform long division. Step one in long division is divide. Well that's not very specific. What do we divide? We take the first term in our dividend and divide it by the first term in our divisor. So we're going to take X squared and divided by X. X squared divided by X is X. So we write that with a degree of one above the term in our dividend that has the same degree so X above the 12 X. And step two in long division is to multiply well what do we multiply? We take the first term in our quotient here and we're going to multiply it by every term in our divisor. So the first term in our quotient is X. And we're going to multiply that by X. X times X is X squared. I'm going to write that under the term with the same degree in our dividend and then X times seven that's plus seven X. Again that goes right under the same term with the same degree from our dividend And step three in long division is to subtract what do we subtract? We subtract the two terms in our dividend. By the two terms from our work that we just did. So we take X squared minus X squared and that is zero. Just as it should be. Those first terms that we're subtracting should be zero. And notice I put parentheses around the two terms in our work in blue that we are subtracting because we want to make sure we subtract all of the terms and not just the first ones. So we take a 12 x minus seven X. And that gives us five X. Okay, that's step three. Step 4 in Long Division. Is to bring down what are we going to bring down? We're going to bring down the term from our dividend. So plus 35 so that now we have the same number of terms in our work. Five X plus 35. That's two terms as we have in our divisor X plus seven. That's two terms. So now we can take starting back over with step number one divide We take the first term in our work five X. And we're going to divide that again by the first term. In our divisor X. Five X divided by X. That is five. So we're going to write plus five up here in our quotient. Step two again is to multiply we're going to take that five and multiply it by both terms in our divisor. So five times X is five X. Five times seven is a positive 35 continuing with step three. We're going to subtract both of these terms here. Five X minus five X. Zero. Perfect. Exactly what we wanted. 35 minus 35. Well, that is also zero when we can't continue with step four. When there's nothing left to bring down than anything we've got here is the remainder and in this case there's a remainder of zero, meaning that our quotient is the answer. We have an answer of X plus five. We look at our answer choices and that matches with answer choice. D. 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). (x^2+8x+15)÷(x+5)

Key Concepts

Polynomial Long Division

Quotient and Remainder

Factoring Polynomials