Sure! To find the expanded form of [tex]\((a - 5)^3\)[/tex], we need to use the binomial theorem. The binomial theorem states that:
[tex]\[
(x + y)^n = \sum_{k=0}^n \binom{n}{k} x^{n-k} y^k
\][/tex]
For the given expression [tex]\((a - 5)^3\)[/tex], we can identify [tex]\(x = a\)[/tex], [tex]\(y = -5\)[/tex], and [tex]\(n = 3\)[/tex]. Let's expand [tex]\((a - 5)^3\)[/tex] using the binomial theorem step-by-step.
### Step-by-Step Expansion:
1. Identify the terms:
[tex]\[
(a - 5)^3
\][/tex]
2. Use the binomial theorem:
[tex]\[
(a - 5)^3 = \sum_{k=0}^3 \binom{3}{k} a^{3-k} (-5)^k
\][/tex]
3. Calculate each term in the sum:
- 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]
4. Combine all the terms:
[tex]\[
(a - 5)^3 = a^3 - 15a^2 + 75a - 125
\][/tex]
So, the expanded form of [tex]\((a - 5)^3\)[/tex] is:
[tex]\[
a^3 - 15a^2 + 75a - 125
\][/tex]
This detailed expansion shows every step required to arrive at the expanded form.
