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Find the product of [tex]\((9c-1)^3\)[/tex].

A. [tex]\(729c^3 - 243c^2 + 27c - 1\)[/tex]
B. [tex]\(729c^3 - 243c^2 + 27c - 3\)[/tex]
C. [tex]\(27c - 3\)[/tex]
D. [tex]\(729c^3 - 1\)[/tex]

Answer :

To find the product [tex]\((9c - 1)^3\)[/tex], we can use the binomial theorem, which provides a way to expand expressions of the form [tex]\((a - b)^3\)[/tex].

The binomial theorem tells us that:

[tex]\[
(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3
\][/tex]

In this expression, [tex]\(a = 9c\)[/tex] and [tex]\(b = 1\)[/tex]. Let's substitute these into the formula and calculate each term step by step:

1. Calculate [tex]\(a^3\)[/tex]:
[tex]\[
(9c)^3 = 9^3 \times c^3 = 729c^3
\][/tex]

2. Calculate [tex]\(-3a^2b\)[/tex]:
[tex]\[
-3 \times (9c)^2 \times 1 = -3 \times 81c^2 = -243c^2
\][/tex]

3. Calculate [tex]\(3ab^2\)[/tex]:
[tex]\[
3 \times 9c \times 1^2 = 27c
\][/tex]

4. Calculate [tex]\(-b^3\)[/tex]:
[tex]\[
-(1)^3 = -1
\][/tex]

Now, we add all these terms together:

[tex]\[
729c^3 - 243c^2 + 27c - 1
\][/tex]

So the fully expanded expression is [tex]\(\boxed{729c^3 - 243c^2 + 27c - 1}\)[/tex].

Referring to the options provided, the correct choice is:

a) [tex]\(729c^3 - 243c^2 + 27c - 1\)[/tex]

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