To factor the expression [tex]\(343 c x^3 - 8 c\)[/tex], we proceed as follows:
1. Identify the Common Factor:
The given expression is [tex]\(343 c x^3 - 8 c\)[/tex]. We can factor out the common factor [tex]\(c\)[/tex] from both terms:
[tex]\[
c(343 x^3 - 8)
\][/tex]
2. Recognize the Difference of Cubes:
The expression inside the parentheses, [tex]\(343 x^3 - 8\)[/tex], is a difference of cubes. Recall that the difference of cubes formula is:
[tex]\[
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
\][/tex]
3. Identify [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
To use this formula, we need to recognize [tex]\(343 x^3\)[/tex] and [tex]\(8\)[/tex] as perfect cubes:
[tex]\[
343 x^3 = (7x)^3 \quad \text{and} \quad 8 = 2^3
\][/tex]
Hence, [tex]\(a = 7x\)[/tex] and [tex]\(b = 2\)[/tex].
4. Apply the Difference of Cubes Formula:
Using the identified values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] in the difference of cubes formula, we get:
[tex]\[
343 x^3 - 8 = (7x)^3 - 2^3 = (7x - 2)\left((7x)^2 + (7x)(2) + 2^2\right)
\][/tex]
5. Compute the Intermediate Values:
Calculate the intermediate expressions:
[tex]\[
(7x)^2 = 49x^2
\][/tex]
[tex]\[
(7x)(2) = 14x
\][/tex]
[tex]\[
2^2 = 4
\][/tex]
6. Combine These into the Factored Form:
Substituting these back in, we get:
[tex]\[
343 x^3 - 8 = (7x - 2)(49x^2 + 14x + 4)
\][/tex]
7. Include the Common Factor c:
Don't forget the factor of [tex]\(c\)[/tex] that we initially factored out. So, the fully factored form is:
[tex]\[
c(343 x^3 - 8) = c(7x - 2)(49x^2 + 14x + 4)
\][/tex]
Thus, the fully factored form of the expression [tex]\(343 c x^3 - 8 c\)[/tex] is:
[tex]\[
\boxed{c(7x - 2)(49x^2 + 14x + 4)}
\][/tex]
