Question
Determine the factors of the expression 8x3−27 .(1 point) Responses (2x−3)(4x2+6x+9) left parenthesis 2 x minus 3 right parenthesis left parenthesis 4 x squared plus 6 x plus 9 right parenthesis (2x+3)(4x2−6x+9) left parenthesis 2 x plus 3 right parenthesis left parenthesis 4 x squared minus 6 x plus 9 right parenthesis (2x−3)(2x2+6x+9) left parenthesis 2 x minus 3 right parenthesis left parenthesis 2 x squared plus 6 x plus 9 right parenthesis (8x−27)(64x2+216x+729)
Answers
Answered by
GPT-4o mini
To factor the expression \(8x^3 - 27\), we can recognize that it is in the form of a difference of cubes. The formula for factoring a difference of cubes \(a^3 - b^3\) is given by:
\[
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
\]
In this case, we can identify:
- \(a = 2x\) (because \((2x)^3 = 8x^3\))
- \(b = 3\) (because \(3^3 = 27\))
Using the formula, we first apply it to get:
1. **Find \(a - b\):**
\[
a - b = 2x - 3
\]
2. **Find \(a^2\), \(ab\), and \(b^2\):**
\[
a^2 = (2x)^2 = 4x^2
\]
\[
ab = (2x)(3) = 6x
\]
\[
b^2 = 3^2 = 9
\]
3. **Combine these to find \(a^2 + ab + b^2\):**
\[
a^2 + ab + b^2 = 4x^2 + 6x + 9
\]
Putting it all together means the factorization of \(8x^3 - 27\) is:
\[
(2x - 3)(4x^2 + 6x + 9)
\]
Thus, the correct response is:
\[
(2x - 3)(4x^2 + 6x + 9)
\]
\[
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
\]
In this case, we can identify:
- \(a = 2x\) (because \((2x)^3 = 8x^3\))
- \(b = 3\) (because \(3^3 = 27\))
Using the formula, we first apply it to get:
1. **Find \(a - b\):**
\[
a - b = 2x - 3
\]
2. **Find \(a^2\), \(ab\), and \(b^2\):**
\[
a^2 = (2x)^2 = 4x^2
\]
\[
ab = (2x)(3) = 6x
\]
\[
b^2 = 3^2 = 9
\]
3. **Combine these to find \(a^2 + ab + b^2\):**
\[
a^2 + ab + b^2 = 4x^2 + 6x + 9
\]
Putting it all together means the factorization of \(8x^3 - 27\) is:
\[
(2x - 3)(4x^2 + 6x + 9)
\]
Thus, the correct response is:
\[
(2x - 3)(4x^2 + 6x + 9)
\]
There are no human answers yet. A form for humans to post answers is coming very soon!