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Algebra (all content)

Course: Algebra (all content)>Unit 10

Lesson 27: Advanced polynomial factorization methods

Factoring sum of cubes

Sal factors 27x^6+125 as (3x^2+5)(9x^4-15x^2+25) using a special product form for a sum of cubes. Created by Sal Khan and Monterey Institute for Technology and Education.

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• Can anyone tell me where to start and end on videos that have to do with Algebra II.
And if there is more than one place to see the Algebra II topics, please list ALL of them.
I'm kind of stuck on topics for the final exam this semester in this class and I can't organize where to watch these videos, so I can fully understand all of Algebra II. I know there is an Algebra I playlist and companion playlists, but I need to know where to get ALL of the Algebra II topics.
• Maybe when you asked that question there wasn't a top level Algebra II group. There is now - just go to https://www.khanacademy.org/math/algebra2
• what if you have a problem that looks like x^3+8 ?
• Then, using the method in this video, that would be (x+2)(x^2-2x+4).
• Thank you! My Algebra 2 teacher is terrible...you have saved my GPA!

Now for my question:

I'm really confused on how you would rewrite the 27x^6. I understand it if there is a number in front and it has an exponent that can be simplified but (for instance) if you had x^3. You don't have a number in front and the exponent can not be simplified. How would I rewrite something like this?

I could really use your help.

~Kelsey
• I believe 27x^6 factored is (3x^2)^3. The coefficient, 27, is a perfect cube of 3 and x^6 is a cube of x^2.
• Why is there a - in front of the parentheses of (3x^2+5)
I believe thats a mistake
• Nope it's an equals sign. At he draws two lines on top of each other, it's pretty messy.
• Why is there a negative sign at ? There shouldn't be a negative sign there.
• how would you solve 64^2-y^3.

My teacher never really went in depth with this process and i'm confused if this can even be solved with the perfect cube patterns?
• What do you mean by solving? What do you have to do? I'm going to assume you have to factor it.

As it happens, 64 = 4^3, that's where you get the cube from.
64^2 = (4^3)^2 = (4^2)^3 = 16^3.

Substituting that gives 64^2 - y^3 = 16^3 - y^3 = (16 - y)(16^2 + 16y + y^2).
• Ive watched this a few times and completely understand
I try and apply it to my math homework but keep getting the wrong answer.
The problem is
x+125x^4
I factor it to x(1+125x^3)
the directions says to use the sum of two cubes but I don't see two cubes and I keep getting the wrong answers
please help! its an online class and I have yet to have my teacher respond to my emailed questions.
• You're partway there.
1+125x^3 can be factored more.
1 is a perfect cube (1 * 1 * 1=1), and so is 125x^3 (5x * 5x * 5x=125x^3)
The formula for the sum of cubes (a^3+b^3) is
a^3+b^3=(a+b)(a^2-ab+b^2)
Plugging in your numbers, we get
(1+5x)(1+5x+25x^2)
Remember the x you factored out.
(x)(5x+1)(25x^2+5x+1)
FYI:
a^3-b^3=(a-b)(a^2+ab+b^2)
a^3+b^3=(a+b)(a^2-ab+b^2)

Hope I could help
• Any tips on remembering where exactly the minus sign goes in the sum of cubes formula? Same with difference of cubes..
(1 vote)
• First, the 2 patterns are very similar. They are identical except for 2 signs.
a^3+b^3 = (a+b)(a^2-ab+b^2)
a^3-b^3 = (a-b)(a^2+ab+b^2)
Notice:
2) The sign on the middle term of the trinomial factor will be the opposite of the sign in the binomial factor.
3) All other signs are positive.

Hope this helps.
• Couldn't you factor the original equation (27x^6+125) as (3x^2+5)^3? Every term is cubed, so why doesn't this work?
(1 vote)
• Exponents don't distribute over addition. Try using the distributive property to multiply together three (3x^2 + 5)s and see that you don't get the desired result.
• how would you factor something with no common variable? For example, (x^3)-3(x^2)+6x-18