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;
- 2×2×2×3 × (x²)³
- Therefore, we have 2³ × 3 × (x²)³
And for -1029y³, we have:
- 7 ×7×7×3 × -y³
- There, we have 7³ × 3 × -y³.
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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