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## Get ready for AP® Calculus

### Course: Get ready for AP® Calculus>Unit 6

Lesson 3: Dividing polynomials by linear factors

# Dividing polynomials by linear expressions

Dividing (3x³+4x²-3x+7) by (x+2) using long division.

## Want to join the conversation?

• Is dividing the polynomials by algebraic long division the only way to solve this problem?
• Well, there is a shortcut called synthetic division that can be used in special cases.
• in why does 2x^2 become 2x
• The steps of polynomial long division are as follows.

1) find the term you have to multiply the leading term of the divisor (denominator) you have to multiply by to get the first term of the dividend (numerator.) In this case the denominator is x+2 and the numerator is 3x^3 + 4x^2 -3x +7. We want what we have to multiply x in x+2 by to get 3x^3. so x times what is 3x^3? the answer is 3x^2. This is the first term of your answer (the quotient.) And just to make sure other terms are explicitly stated, the dividend is 3x^3 + 4x^2 -3x +7 and divisor is x + 2

2) take the term of the quotient you just found and multiply it by the divisor. so the term found was 3x^2 and the divisor is x + 2, so ultiply those to get 3x^3 + 6x^2. You should notice 3x^3 is the same as the first term in the dividend.

3) subtract this new expression from the dividend. so (3x^3 + 4x^2 -3x +7) - (3x^3 + 6x^2) = -2x^2 - 3x + 7. THIS is your new divisor, or in other words you can think of the division problem now as (-2x^2 - 3x + 7) / (x + 2).

4) Repeat steps 1-3 to find the next term of your quotient, the expression you subtract from the current dividend and then the new dividend.

so step 1, what do we multiply the first term of the divisor by to get the first term of the dividend? -2x (this is where the -2x you asked about came from.) So the second term of the quotient is -2x, Then you multiply by the dividend -2x(x + 2) = -2x^2 - 4x. now subtract this from the current dividend (-2x^2 - 3x + 7) - (-2x^2 - 4x) = x + 7.

Now just keep going until the dividend has a smaller degree than the divisor. Then this is your remainder.

Let me know if that didn't help. just to make sure your question specifically is answered, the -2x is the number that multiplies the first term of the divisor to get the first term of the new dividend, which is part of the process of solving these problems.
• What happened to the synthetic division video and practice?
• They were changes on this site so I saw, so I guess they changed a few things.
• Do we still have to put a condition on the domain that 'x cannot equal -2', even though the remainder of 'x + 2' clearly shows that a value of -2 would create an undefined denominator?
• There are many ways to write the domain, and it really depends on your teacher. For the easiest, you may write (just going off you're example), (-inf,inf) x ≠ -2.

More formally, which would be nice, is to use the "union term." More specifically, instead of (-inf,inf) x ≠ -2, it would be (-inf,-2]U[-2,inf). Why union? It's the term for the combination of two unconnected intervals. Hopefully this helps.
• I have a question. When Khan does 4 - (-6) he gets 2. However, in practice, I have to make the 6 positive and get 10. Why doesn't Khan fix the problem?
• Sal is not actually doing 4 - (-6) but is actually doing 4 - 6, which becomes negative 2.
• In Hungary, we do not learn (algebraic) long division, therefore I do not understand what Sal is doing in the video. As far as I am concerned, dividing polynomials by linear expressions is needed for some integration problems, so could you recommend me a video in connection with the topic, or suggest another method?
• If you look under the Arithmetic section in Khan Academy, then go to Multiplication and Division, there are videos on how to perform long division.
• in the past video we would stop and end it as a remainder in ,, why are we continuing? or the question is when do I know to stop
• At , you still have an x on the bottom, so you have not reached the remainder part yet. He finally gets the remainder of 5 at . This is also clearly seen when he rewrites at .
• i no understand
• Why do we always focus on the highest degree term? Can't we start dividing by the constant?
• We can. Try it. Dividing by the highest degree term is simply convention.
(1 vote)
• We Know Dividend = Divisor * Quotient + Reminder
So 3x^3+4x^2-3x+7 Should Be Written As (x+2)(3x^2-2x+1)+5,
Buy Why It Is Written As 3x^2-2x+1+(5/x+2)?