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### Course: Algebra 2>Unit 4

Lesson 3: Dividing polynomials by linear factors

# Factoring using polynomial division

I'm here to help you teach this video to your students. Here are a few ideas for things you can ask me: Rate this response Identify the main points of the video, and summarize them in 40-60 words at a 6th grade level. Optimize the text for SEO. Don't use the words "in this video" - just get to the point. Use active voice. Avoid the words magic, adventure, dive, lowdown, fun, and world. The video breaks down the process of dividing polynomials by linear factors. It starts with a given polynomial and a known factor, then uses polynomial division to rewrite the expression as a product of linear factors. The video emphasizes understanding the steps and the reasoning behind each one.

## Want to join the conversation?

• i dont understand i think my brain fried, i have watched this video more than 3 times i am a lost boy in neverland.
• Lena, try rewatch the video again, and pause it when you don't understand, and rewatch that part again.
Hope this helps :D
• At , he says the polynomial wouldn't be so easy to factor, so my question is how do you factor this polynomial if you don't have the given linear factor.
• Well, you have two real choices. You can factor by grouping:

Another method is called the "rational root theorem"... which I wasn't able to find on this site. If someone else finds it, please link it. It's a pain to use, and it doesn't work to find irrational or imaginary roots... but it does find all of the rational roots (plus a couple extra)... Here's how it works:

For the equation: 4x^3 + 19x^2 + 19x - 6, take the last coefficient, and divide it by the lead coefficient.

6/4

Try all possible combinations of the factors of the numerator, with all possible factors of the denominator:

6/4, 3/4, 2/4, 1/4, 6/2, 3/2, 2/2, 1/2, 6, 3, 2, 1

OR: 3/2, 3/4, 1/2, 1/4, 3, 1, 6, 2
OR (in order): 6, 3, 2, 3/2, 1, 3/4, 1/2, 1/4

Then divide the polynomial by x +- (each one of those numbers listed above). When you're done dividing, if you don't get a remainder, then congratulations! You found a root! If you get a remainder, then it wasn't really a root.

Pro-tip: If you do use this method, learn synthetic division, as it really speeds up all the dividing you have to do:

With any luck, you'll be able to factor by grouping.
• what i there is a remainder?
• Winding up with a remainder is just that. Theres no way to make it a "nice" polynimial. It'd basically be a polynomial plus some rational function.

A quick example, I'm just making these up, (x^4+2x^3-3x^2+5x-7)/(x-2) would get you x^3+4x^2+5x+15 + 23/(x-2). the 23/(x-2) is that rational function I mentioned. Hopefully you notice x-2 is in the denominator and is the divisor int he division problem. This is always the case, and the numerator is at most one degree less than the denominator.
• At he says linear factors, but doesn't explain what that exactly means....
• "Linear factors" is just a phrase for a factor that looks like (ax+b).

At the end of the video we see the factors are (x+2), (x+3), and (4x-1), which all follow that format.
• math is beautiful
• Math is the best.
(1 vote)
• How do you get the 12 from the equation?
• By multiplying a*c of the quadratic 4x^2+11x-3 and then finding factors of a*c that when added together gets you +11, which happens to be +12 and -1. That’s how you would break up +11, then start grouping, and go from there to factor. Hope that answers your question.
• I'm sorry, but I'm still confused about how you break down the equation after finding the quadratic for the polynomial that you get after dividing.
• At about , he says about that you could use the quadratic formula to factor into a linear expression if the quadratic was hard to factor, so my question is how do you factor this polynomial if it's hard to factor using the quadratic formula to put it in linear factors?
• What happened to the video description?