9x^2 6xy 4y^2 is a factor of 27x^3 - 8y^3, find the other factor.
27x^3-8y^3 is the difference of two cubes: (3x)^3 - (2y)^3
So, it factors as (3x-2y)(9x^2+6xy+4y^2)
To find the other factor, we can use polynomial division. Here's how you can do it:
1. Set up the long division, with the divisor (9x^2 + 6xy + 4y^2) on the left and the dividend (27x^3 - 8y^3) on the right:
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9x^2 + 6xy + 4y^2 | 27x^3 + 0x^2 + 0xy - 8y^3
2. Divide the first term of the dividend (27x^3) by the first term of the divisor (9x^2). This gives us 3x:
3x
3. Now, multiply the divisor (9x^2 + 6xy + 4y^2) by the result from step 2 (3x), and write the product under the dividend, aligned with the appropriate terms:
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9x^2 + 6xy + 4y^2 | 27x^3 + 0x^2 + 0xy - 8y^3
-(27x^3 + 18x^2y + 12xy^2)
4. Subtract this product from the dividend. Since we have a negative term, we can rewrite the subtraction as addition:
3x
+(0x^2 + 12xy^2)
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9x^2 + 6xy + 4y^2 | 27x^3 + 0x^2 + 0xy - 8y^3
-(27x^3 + 18x^2y + 12xy^2)
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-18x^2y - 12xy^2
5. Now, bring down the next term from the dividend (0xy):
3x
+(0x^2 + 12xy^2)
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9x^2 + 6xy + 4y^2 | 27x^3 + 0x^2 + 0xy - 8y^3
-(27x^3 + 18x^2y + 12xy^2)
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-18x^2y - 12xy^2 + 0xy
6. Divide the first term of this new expression (-18x^2y) by the first term of the divisor (9x^2). This gives us -2y:
3x - 2y
7. Multiply the divisor (9x^2 + 6xy + 4y^2) by the result from step 6 (-2y):
3x - 2y
-(18x^2y - 12xy^2)
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9x^2 + 6xy + 4y^2 | 27x^3 + 0x^2 + 0xy - 8y^3
-(27x^3 + 18x^2y + 12xy^2)
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-18x^2y - 12xy^2 + 0xy
+(18x^2y - 12xy^2)
8. Subtract this product from the new expression:
3x - 2y
-(18x^2y - 12xy^2)
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9x^2 + 6xy + 4y^2 | 27x^3 + 0x^2 + 0xy - 8y^3
-(27x^3 + 18x^2y + 12xy^2)
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-18x^2y - 12xy^2 + 0xy
+(18x^2y - 12xy^2)
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0xy
9. We have reached a remainder of 0xy, indicating that we have successfully divided the polynomial. The other factor is the quotient we obtained, which is 3x - 2y.
Therefore, the other factor is 3x - 2y