Question

Which of the following is a factor of 24x6 − 1029y3?

24
2x2 + 7y
4x4 + 14x2y + 49y2
All of the above

Answer 1

`4x^4+14x^2y+49y^2`

Explanation:

Given expression:

`24x^6-1029y^3`

Factor out the common term 3:

`\implies 3(8x^6-343y^3)`

Rewrite 8 as 2³ and 343 as 7³:

`\implies 3(2^3x^6-7^3y^3)`

`\textsf{Rewrite $x^6$ as $x^(2 \cdot 3)$}:`

`\implies 3(2^3x^(2 \cdot 3)-7^3y^3)`

`\textsf{Apply the exponent rule} \quad a^(bc)=(a^b)^c:`

`\implies 3(2^3(x^2)^3-7^3y^3)`

`\textsf{Apply the exponent rule} \quad a^cb^c=(ab)^c:`

`\implies 3((2x^2)^3-(7y)^3)`

Apply the Difference of Cubes formula a³ - b³ = (a - b)(a² + ab + b²):

`\implies 3(2x^2-7y)((2x^2)^2+(2x^2)(7y)+(7y)^2)`

`\textsf{Apply the exponent rule} \quad (ab)^c=a^cb^c:`

`\implies 3(2x^2-7y)(2^2x^4+2x^27y+7^2y^2)`

`\implies 3(2x^2-7y)(4x^4+14x^2y+49y^2)`

Therefore, the only answer option that is a factor of the given expression is:

`\boxed{4x^4+14x^2y+49y^2}`

Answer 2

• C) 4x⁴ + 14x²y + 49y²

Explanation:

Given expression

• 24x⁶ - 1029y³

Factorize it as follows:

• 24x⁶ - 1029y³ =
• 3(8x⁶ - 343y³) =
• 3[(2x²)³ - (7y)³] =
• 3(2x² - 7y)[(2x²)² + (2x²)(7y) + (7y)²] =
• 3(2x² - 7y)(4x⁴ + 14x²y + 49y²)

Used identity:

• a³ - b³ = (a - b)(a² + ab + b²)

Correct choice is C