Sure, let's solve the given problem step-by-step:
### Step 1: Identify the Dividend and Divisor
We are given:
- Dividend: \( x^3 - 216a^3 \)
- Divisor: \( x - 6a \)
### Step 2: Perform Polynomial Division
We need to divide \( x^3 - 216a^3 \) by \( x - 6a \). To do so, we will use polynomial long division.
1. First term of the quotient:
- Divide the leading term of the dividend \( x^3 \) by the leading term of the divisor \( x \) to get: \( \frac{x^3}{x} = x^2 \).
2. Subtract the result from the dividend:
- Multiply \( x^2 \) by the divisor \( x - 6a \):
[tex]\[
x^2 \cdot (x - 6a) = x^3 - 6ax^2
\][/tex]
- Subtract this from the dividend:
[tex]\[
(x^3 - 216a^3) - (x^3 - 6ax^2) = -6ax^2 - 216a^3
\][/tex]
3. Second term of the quotient:
- Divide the new leading term \( -6ax^2 \) by the leading term of the divisor \( x \) to get: \( \frac{-6ax^2}{x} = -6ax \).
4. Subtract the result from the intermediate dividend:
- Multiply \( -6ax \) by the divisor \( x - 6a \):
[tex]\[
-6ax \cdot (x - 6a) = -6ax^2 + 36a^2x
\][/tex]
- Subtract this from the intermediate dividend:
[tex]\[
(-6ax^2 - 216a^3) - (-6ax^2 + 36a^2x) = -36a^2x - 216a^3
\][/tex]
5. Third term of the quotient:
- Divide the new leading term \( -36a^2x \) by the leading term of the divisor \( x \) to get: \( \frac{-36a^2x}{x} = -36a^2 \).
6. Subtract the result from the intermediate dividend:
- Multiply \( -36a^2x \) by the divisor \( x - 6a \):
[tex]\[
36a^2 \cdot (x - 6a) = 36a^2x - 216a^3
\][/tex]
- Subtract this from the intermediate dividend:
[tex]\[
(-36a^2x - 216a^3) - (36a^2x - 216a^3) = 0
\][/tex]
### Step 3: Determine the Quotient and Remainder
After performing the division, we find:
- Quotient: \( 36a^2 + 6ax + x^2 \)
- Remainder: \( 0 \)
### Step 4: Verification
To verify, we check that \( \text{Quotient} \cdot \text{Divisor} + \text{Remainder} = \text{Dividend} \):
Calculate:
[tex]\[
(x - 6a)(36a^2 + 6ax + x^2) + 0 = x^3 - 216a^3
\][/tex]
Expanding \( (x - 6a)(36a^2 + 6ax + x^2) \):
[tex]\[
(x - 6a)(36a^2 + 6ax + x^2) = x \cdot 36a^2 + x \cdot 6ax + x \cdot x^2 - 6a \cdot 36a^2 - 6a \cdot 6ax - 6a \cdot x^2
\][/tex]
Simplify:
[tex]\[
= 36a^2 x + 6a x^2 + x^3 - 216a^3 - 36a^2 x - 6a x^2
= x^3 - 216a^3
\][/tex]
The expression simplifies and confirms our calculation.
### Conclusion
- The quotient is \( 36a^2 + 6ax + x^2 \).
- The remainder is \( 0 \).
Thus, we have verified that the quotient and remainder calculation is correct, satisfying [tex]\( (36a^2 + 6ax + x^2)(x - 6a) + 0 = x^3 - 216a^3 \)[/tex].
