Question
determine the factors of the expression 8x^3-27
Answers
Answered by
GPT-4o mini
To factor the expression \( 8x^3 - 27 \), we recognize that it can be expressed as a difference of cubes. The expression can be rewritten as:
\[
(2x)^3 - 3^3
\]
This follows the formula for factoring a difference of cubes, which is:
\[
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
\]
In our case, \( a = 2x \) and \( b = 3 \). Using the formula:
1. \( a - b = 2x - 3 \)
2. \( a^2 = (2x)^2 = 4x^2 \)
3. \( ab = (2x)(3) = 6x \)
4. \( b^2 = 3^2 = 9 \)
Putting this all together, we have:
\[
a^3 - b^3 = (2x - 3)(4x^2 + 6x + 9)
\]
Thus, the factored form of \( 8x^3 - 27 \) is:
\[
(2x - 3)(4x^2 + 6x + 9)
\]
\[
(2x)^3 - 3^3
\]
This follows the formula for factoring a difference of cubes, which is:
\[
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
\]
In our case, \( a = 2x \) and \( b = 3 \). Using the formula:
1. \( a - b = 2x - 3 \)
2. \( a^2 = (2x)^2 = 4x^2 \)
3. \( ab = (2x)(3) = 6x \)
4. \( b^2 = 3^2 = 9 \)
Putting this all together, we have:
\[
a^3 - b^3 = (2x - 3)(4x^2 + 6x + 9)
\]
Thus, the factored form of \( 8x^3 - 27 \) is:
\[
(2x - 3)(4x^2 + 6x + 9)
\]
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