- #1
MorallyObtuse
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Homework Statement
Express as the product of four factors
Is this correct?
[tex]a^6 - b^6 = (a - b)(a^5 + a^4b + a^3b^2 + a^2b^3 + ab^4 + b^3)[/tex]
In summary: Then you should talk to your teacher about it.I did get it, just joking with 1/2" :biggrin:And thanks very much :)
Express as the product of four factors
Is this correct?
[tex]a^6 - b^6 = (a - b)(a^5 + a^4b + a^3b^2 + a^2b^3 + ab^4 + b^3)[/tex]
- #2
HallsofIvy
Science Advisor
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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 [itex]a^6- b^6[/itex] as [itex](a^3)^2- (a^3)^2[/itex].MorallyObtuse said:
- #3
MorallyObtuse
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I don't get it?!
- #4
1/2"
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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.
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
MorallyObtuse
- 45
- 0
1/2" said:
No I don't get it
- #6
Mark44
Mentor
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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.
Each factor on the right can be further factored using known formulas for the sum of cubes and difference of cubes.
- #7
HallsofIvy
Science Advisor
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MorallyObtuse said:
Then you should talk to your teacher about it.MorallyObtuse said:No I don't get it
- #8
MorallyObtuse
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- 0
I did get it, just joking with 1/2"
- #9
MorallyObtuse
- 45
- 0
And thanks very much :)
[itex](a-b) (a^2+ab+b^2) (a+b)(a^2-ab+b^2)[/itex]
[itex](a-b) (a^2+ab+b^2) (a+b)(a^2-ab+b^2)[/itex]
- #10
Borek
Mentor
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[tex]1 \times 1 \times 1 \times (a^6 - b^6)[/tex]
- #11
Mark44
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Heck, I can find 7 factors!
[tex]1 \times 1 \times 1 \times 1 \times 1 \times 1 \times (a^6 - b^6)[/tex]
[tex]1 \times 1 \times 1 \times 1 \times 1 \times 1 \times (a^6 - b^6)[/tex]
- #12
HallsofIvy
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Yes, but that would be marked wrong because the problem asked for four factors. Borek's brilliant answer gets the gold star!
- #13
Mark44
Mentor
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I was hoping for extra credit because I went above and beyond the requirements.
FAQ: Express as the product of four factors
1. What does it mean to express a number as the product of four factors?
When we express a number as the product of four factors, we are breaking down the number into four smaller numbers that, when multiplied together, equal the original number. This can also be thought of as finding the prime factorization of a number.
2. Why is it important to express a number as the product of four factors?
Expressing a number as the product of four factors can help us understand the factors and factors of factors that make up a number. This can be useful in many areas of math, such as finding common factors or simplifying fractions.
3. How do you find the factors of a number?
To find the factors of a number, we can start by listing all of the possible factors and then narrowing down the list by testing each factor to see if it divides evenly into the original number. Another method is to use prime factorization, where we find the prime factors of a number and then combine them to find all of the factors.
4. Can every number be expressed as the product of four factors?
Yes, every number can be expressed as the product of four factors. This is because every number can be broken down into prime factors, and we can combine these prime factors to create four factors that equal the original number.
5. How can we use the product of four factors to solve equations?
We can use the product of four factors to solve equations by breaking down the numbers in the equation into their prime factors and then rearranging them to find the solution. This can be a useful method for solving equations with large numbers or multiple variables.