Dividing polynomials
-
Polynomial Division
-
Polynomial divided by monomial
-
Dividing multivariable polynomial with monomial
-
Dividing polynomials 1
-
Dividing polynomials with remainders
-
Synthetic Division
-
Synthetic Division Example 2
-
Why Synthetic Division Works
-
Factoring Sum of Cubes
-
Difference of Cubes Factoring
Factoring Sum of Cubes Factoring Sum of Cubes
⇐ Use this menu to view and help create subtitles for this video in many different languages.
You'll probably want to hide YouTube's captions if using these subtitles.
- Factor twenty seven x to the sixth plus one twenty five
- So this is a pretty interesting problem and frankly
- the only way to do this
- is if you recognize it as a special form
- and what I want to do
- is kind of show you the special form right first
- and then we can kind of pattern match.
- So the special form is
- if I were to take
- and this is really something you need to know
- and I'd argue whether you really need to know this
- but actually to do this problem
- it's something you need to know
- and that's if you have
- a squared minus a b plus b squared
- and you multiply that times a plus b
- let's think about what we're going to get
- So we're gonna take a product right here
- we're multiplying
- so let's do some algebraic multiplication
- so let's multiply b times b squared is b to the third
- b times negative a b is negative a b squared
- b times a squared is a squared b
- Now let's multiply this top term times a
- a times b squared is a b squared
- a times negative a b is negative a squared b
- and then a times a squared is a to the third
- and then we just have to add up all of the terms
- We have a negative a squared b -
- Oh, we have a positive a squared b
- and a negative a squared b
- so these guys cancel out
- we have a negative a b squared
- and a positive a b squared
- These guys cancel out.
- So all we're left with is
- an a to the third here
- a to the third
- and then plus, plus this b to the third
- plus this b to the third
- Or another way to think about it
- if someone gives you a to the third
- plus b to the third
- this can be factored into these two expressions
- That can be factored into
- a plus b times
- a squared minus a b plus b squared
- So this is essentially the special form
- If you have a sum of cubes
- it can be factored out as the sum of the cube roots
- plus- er, the sum of the cube roots times
- this expression right here
- and we just showed that it works
- So let's see if we have that special form here
- Well 27 is definitely the cube of three
- three to the third power is twenty seven
- x to the sixth is also the cube of x squared
- if you raise x to the sixth to the one third power
- you get x squared
- So this first term right over here can be rewritten
- as three x squared to the third power
- and the second term right here-
- that's five to the third power
- so plus five to the third power
- And this might be a little bit confusing for you
- so just let's - never hurts to review
- let's multiply three x squared times
- three x squared times three x squared
- that is literally equal to three times three
- times three times x squared times x squared
- times x squared
- that's this part right here is twenty seven
- x squared times x squared is x to the fourth
- times x squared is x to the sixth
- Or you could just raise both of these to
- the third power
- three to the third is twenty seven
- x squared to the third power -
- you take an exponent to an exponent
- and you're gonna take the product
- of the exponents
- so it'll be x to the two times three or
- x to the sixth power
- So now we know that we have this pattern
- So we can just use this
- We have the sum of cubes
- So just by using this pattern right over here
- that means that we can factor it as
- this is going to be equal to
- three x squared
- that's our a let me make it clear
- this right here is our a
- this right here is our b
- so it's going to be a plus b
- so it's going to be three x squared plus
- b
- plus five times a squared
- a squared
- let me do this in a new color
- so three x squared squared
- let's start thinking about that for a second
- three x squared squared
- well that's going to be nine
- x to the fourth
- so it's going to be times nine x to the fourth
- minus the product of these two things
- so minus the product of five and three x squared
- so minus fifteen x squared
- and then finally plus b squared
- b is five
- so it's going to be five squared
- so plus twenty five
- when I said b is
- this is b - not the whole five to the third
- and when I say a just this part is a
- And we're done
- and we could
- I won't explain it in detail in this video
- but this right here
- if we're thinking about real numbers
- we can't actually factor this any more
- so we are done factoring this
- And remember this is really just a very
- very very special case of being able to recognize
- the sum of cubes
Be specific, and indicate a time in the video:
At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
|
Have something that's not a question about this content? |
This discussion area is not meant for answering homework questions.
Discuss the site
For general discussions about Khan Academy, visit our Reddit discussion page.
Flag inappropriate posts
Here are posts to avoid making. If you do encounter them, flag them for attention from our Guardians.
abuse
- disrespectful or offensive
- an advertisement
not helpful
- low quality
- not about the video topic
- soliciting votes or seeking badges
- a homework question
- a duplicate answer
- repeatedly making the same post
wrong category
- a tip or feedback in Questions
- a question in Tips & Feedback
- an answer that should be its own question
about the site
Share a tip
Suggest a fix
Have something that's not a tip or feedback about this content?
This discussion area is not meant for answering homework questions.