Express as the product of four factors

1. Nov 23, 2009

MorallyObtuse

1. The problem statement, all variables and given/known data
Express as the product of four factors
Is this correct?
$$a^6 - b^6 = (a - b)(a^5 + a^4b + a^3b^2 + a^2b^3 + ab^4 + b^3)$$

2. Nov 23, 2009

HallsofIvy

Staff Emeritus
What you have written is true. It is not a "correct" response to the problem because, obviously, it does not have four factors. Start again. Think of $a^6- b^6$ as $(a^3)^2- (a^3)^2$.

3. Nov 23, 2009

MorallyObtuse

I don't get it?!

4. Nov 23, 2009

1/2"

Hi MorallyObtuse,
By the law of indices (a m)n=amxn
and so you can also represent it like this(like HallsofIvy said)
=(a^3)^2-(b^3)^2
And then you can simplyfy it like this
=(a^3-b^3)(a^3+b^3)
and finaly u have your 4 terms
(a-b) (a^2+ab+b^2) (a+b)(a^2-ab+b^2)
I think you get it.

5. Nov 23, 2009

MorallyObtuse

No I don't get it

6. Nov 23, 2009

Staff: Mentor

a6 - b6 = (a3 + b3)(a3 - b3)
Each factor on the right can be further factored using known formulas for the sum of cubes and difference of cubes.

7. Nov 24, 2009

HallsofIvy

Staff Emeritus
Then you should talk to your teacher about it.

8. Nov 24, 2009

MorallyObtuse

I did get it, just joking with 1/2"

9. Nov 24, 2009

MorallyObtuse

And thanks very much :)

$(a-b) (a^2+ab+b^2) (a+b)(a^2-ab+b^2)$

10. Nov 24, 2009

Staff: Mentor

$$1 \times 1 \times 1 \times (a^6 - b^6)$$

11. Nov 24, 2009

Staff: Mentor

Heck, I can find 7 factors!
$$1 \times 1 \times 1 \times 1 \times 1 \times 1 \times (a^6 - b^6)$$

12. Nov 24, 2009

HallsofIvy

Staff Emeritus
Yes, but that would be marked wrong because the problem asked for four factors. Borek's brilliant answer gets the gold star!

13. Nov 25, 2009

Staff: Mentor

I was hoping for extra credit because I went above and beyond the requirements.

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