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Factor twenty seven x to the sixth plus one twenty five

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So this is a pretty interesting problem and frankly

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the only way to do this

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is if you recognize it as a special form

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and what I want to do

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is kind of show you the special form right first

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and then we can kind of pattern match.

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So the special form is

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if I were to take

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and this is really something you need to know

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and I'd argue whether you really need to know this

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but actually to do this problem

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it's something you need to know

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and that's if you have

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a squared minus a b plus b squared

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and you multiply that times a plus b

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let's think about what we're going to get

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So we're gonna take a product right here

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we're multiplying

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so let's do some algebraic multiplication

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so let's multiply b times b squared is b to the third

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b times negative a b is negative a b squared

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b times a squared is a squared b

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Now let's multiply this top term times a

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a times b squared is a b squared

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a times negative a b is negative a squared b

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and then a times a squared is a to the third

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and then we just have to add up all of the terms

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We have a negative a squared b -

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Oh, we have a positive a squared b

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and a negative a squared b

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so these guys cancel out

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we have a negative a b squared

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and a positive a b squared

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These guys cancel out.

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So all we're left with is

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an a to the third here

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a to the third

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and then plus, plus this b to the third

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plus this b to the third

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Or another way to think about it

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if someone gives you a to the third

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plus b to the third

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this can be factored into these two expressions

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That can be factored into

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a plus b times

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a squared minus a b plus b squared

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So this is essentially the special form

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If you have a sum of cubes

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it can be factored out as the sum of the cube roots

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plus- er, the sum of the cube roots times

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this expression right here

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and we just showed that it works

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So let's see if we have that special form here

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Well 27 is definitely the cube of three

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three to the third power is twenty seven

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x to the sixth is also the cube of x squared

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if you raise x to the sixth to the one third power

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you get x squared

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So this first term right over here can be rewritten

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as three x squared to the third power

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and the second term right here-

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that's five to the third power

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so plus five to the third power

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And this might be a little bit confusing for you

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so just let's - never hurts to review

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let's multiply three x squared times

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three x squared times three x squared

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that is literally equal to three times three

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times three times x squared times x squared

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times x squared

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that's this part right here is twenty seven

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x squared times x squared is x to the fourth

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times x squared is x to the sixth

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Or you could just raise both of these to

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the third power

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three to the third is twenty seven

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x squared to the third power -

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you take an exponent to an exponent

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and you're gonna take the product

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of the exponents

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so it'll be x to the two times three or

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x to the sixth power

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So now we know that we have this pattern

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So we can just use this

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We have the sum of cubes

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So just by using this pattern right over here

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that means that we can factor it as

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this is going to be equal to

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three x squared

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that's our a let me make it clear

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this right here is our a

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this right here is our b

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so it's going to be a plus b

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so it's going to be three x squared plus

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b

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plus five times a squared

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a squared

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let me do this in a new color

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so three x squared squared

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let's start thinking about that for a second

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three x squared squared

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well that's going to be nine

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x to the fourth

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so it's going to be times nine x to the fourth

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minus the product of these two things

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so minus the product of five and three x squared

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so minus fifteen x squared

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and then finally plus b squared

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b is five

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so it's going to be five squared

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so plus twenty five

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when I said b is

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this is b - not the whole five to the third

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and when I say a just this part is a

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And we're done

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and we could

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I won't explain it in detail in this video

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but this right here

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if we're thinking about real numbers

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we can't actually factor this any more

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so we are done factoring this

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And remember this is really just a very

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very very special case of being able to recognize

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the sum of cubes