Which of the following is a factor of 24x6 − 1029y3?
24
2x2 + 7y
4x4 + 14x2y + 49y2
all of the above

Here, we are required to find which of the following is a factor of 24x⁶ - 1029y³.

The correct factors are ;

2x2 + 7y

2x2 + 7y 4x4 + 14x2y + 49y2

To find the factors of 24x⁶ - 1029y³, we need to first find the prime factors of each term.

Therefore, for 24x⁶, we have;

And for -1029y³, we have:

Therefore, since 24 is not a common factor for both terms, we can conclude that it is not a factor of 24x⁶ - 1029y³.

Moving on, since 3 is a factor, 24x⁶ - 1029y³ then becomes;

And since, 2x² and -7y are the cubic roots of 8x⁶ and -343y³ respectively.

Therefore, 2x² - 7y is a factor.

Consequently, the division of 8x⁶ -343y³ yields;

4x⁴ + 14x²y + 49 which is a factor.

However, further factorisation of 4x⁴ + 14x²y + 49 yields (2x² + 7y)(2x² + 7y).

From above, we can conclude that (2x² + 7y) and 4x⁴ + 14x²y + 49 are both factors of 24x⁶ - 1029y³.

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The answer to your question is only the second and the third are factors.

Step-by-step explanation:

Find the factors of               24x⁶ − 1029y³

First find the prime factors of 24 and 1029

  24  2                1029 3

  12   2                  343 7                  Then 24 = 2³3

   6   2                    49  7                         1029 = 7³ 3

   3   3                       7  7

   1                             1

     x⁶ = (x²)³                  y³ = y³

Then

                                  2³3(x²)³  -  7³ 3  y³

Factor 3                    3[  2³(x²)³  -  7³ y³]

Factor                      3 [ (2x² - 7y)(4x⁴ + 14x²y + 49y²)]

Then

24

2x2 + 7y

4x4 + 14x2y + 49y2

All of the above