To factor the expression [tex]\( b^3 - 1000 \)[/tex], let's follow these detailed steps:
1. Recognize the Pattern:
We notice that [tex]\( b^3 - 1000 \)[/tex] is a difference of cubes. The general formula for the difference of cubes is:
[tex]\[
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
\][/tex]
2. Identify [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
In our case, [tex]\( a^3 = b^3 \)[/tex] because we start with [tex]\( b^3 \)[/tex]. For [tex]\(1000\)[/tex], we recognize [tex]\( 1000 = 10^3 \)[/tex]. Thus, we can set:
[tex]\[
a = b \quad \text{and} \quad b = 10
\][/tex]
3. Apply the Difference of Cubes Formula:
Substituting [tex]\(a\)[/tex] and [tex]\(b\)[/tex] into the formula we get:
[tex]\[
b^3 - 10^3 = (b - 10)\left(b^2 + b \cdot 10 + 10^2\right)
\][/tex]
4. Simplify the Expression:
Let's simplify the factors within the parentheses:
[tex]\[
10^2 = 100
\][/tex]
[tex]\[
b \cdot 10 = 10b
\][/tex]
Hence, our expression within the parentheses becomes:
[tex]\[
b^2 + 10b + 100
\][/tex]
5. Form the Factored Expression:
The factored form of [tex]\( b^3 - 1000 \)[/tex] is:
[tex]\[
(b - 10)(b^2 + 10b + 100)
\][/tex]
6. Identify the Correct Answer:
Comparing with the provided options:
[tex]\[
\text{B. } (b - 10)(b^2 + 10b + 100)
\][/tex]
Therefore, the correct factored form is option B: [tex]\((b - 10)(b^2 + 10b + 100)\)[/tex].
