Discover a world of knowledge at Westonci.ca, where experts and enthusiasts come together to answer your questions. Join our Q&A platform and get accurate answers to all your questions from professionals across multiple disciplines. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts.
Sagot :
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].
### 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].
We hope our answers were useful. Return anytime for more information and answers to any other questions you have. We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. Thank you for trusting Westonci.ca. Don't forget to revisit us for more accurate and insightful answers.
