Sure, let’s break down the process of expanding the expression [tex]\((a - 5)^3\)[/tex] step-by-step:
1. Understand the Binomial Theorem: The binomial theorem states that [tex]\((x + y)^n\)[/tex] can be expanded as the sum of terms of the form [tex]\(\binom{n}{k} x^{n-k} y^k\)[/tex], where [tex]\(\binom{n}{k}\)[/tex] is a binomial coefficient.
2. Identify the Components: For our expression [tex]\((a - 5)^3\)[/tex]:
- [tex]\(x\)[/tex] is [tex]\(a\)[/tex]
- [tex]\(y\)[/tex] is [tex]\(-5\)[/tex]
- [tex]\(n\)[/tex] is [tex]\(3\)[/tex]
3. Apply the Binomial Theorem:
[tex]\[
(a - 5)^3 = \sum_{k=0}^{3} \binom{3}{k} a^{3-k} (-5)^k
\][/tex]
4. Calculate Each Term:
- For [tex]\(k = 0\)[/tex]:
[tex]\[
\binom{3}{0} a^{3-0} (-5)^0 = 1 \cdot a^3 \cdot 1 = a^3
\][/tex]
- For [tex]\(k = 1\)[/tex]:
[tex]\[
\binom{3}{1} a^{3-1} (-5)^1 = 3 \cdot a^2 \cdot -5 = -15a^2
\][/tex]
- For [tex]\(k = 2\)[/tex]:
[tex]\[
\binom{3}{2} a^{3-2} (-5)^2 = 3 \cdot a^1 \cdot 25 = 75a
\][/tex]
- For [tex]\(k = 3\)[/tex]:
[tex]\[
\binom{3}{3} a^{3-3} (-5)^3 = 1 \cdot a^0 \cdot -125 = -125
\][/tex]
5. Sum the Terms:
[tex]\[
a^3 - 15a^2 + 75a - 125
\][/tex]
Therefore, the expanded form of [tex]\((a - 5)^3\)[/tex] is:
[tex]\[
a^3 - 15a^2 + 75a - 125
\][/tex]
