Let's perform the long division of the polynomial \(12x^3 + 8x^2 - 7x - 912\) by the polynomial \(3x - 13\).
Step 1: Set up the long division.
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3x - 13 | 12x^3 + 8x^2 - 7x - 912
Step 2: Divide the leading term of the dividend by the leading term of the divisor.
- Divide \(12x^3\) by \(3x\):
\[ \frac{12x^3}{3x} = 4x^2 \]
Write \(4x^2\) above the division line.
4x^2
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3x - 13 | 12x^3 + 8x^2 - 7x - 912
Step 3: Multiply the entire divisor \(3x - 13\) by \(4x^2\) and write the result below the dividend.
\[ 4x^2 \cdot (3x - 13) = 12x^3 - 52x^2 \]
Now write this below the corresponding terms in the dividend.
4x^2
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3x - 13 | 12x^3 + 8x^2 - 7x - 912
- (12x^3 - 52x^2)
_____________________
60x^2 - 7x - 912
Step 4: Subtract this from the dividend.
60x^2 - 7x - 912
Step 5: Bring down the next term (here you already have it).
Step 6: Divide again. Divide \(60x^2\) by \(3x\):
\[ \frac{60x^2}{3x} = 20x \]
Write \(20x\) above the division line next to \(4x^2\).
4x^2 + 20x
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3x - 13 | 12x^3 + 8x^2 - 7x - 912
- (12x^3 - 52x^2)
_____________________
60x^2 - 7x - 912
- (60x^2 - 260x)
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253x - 912
Step 7: Subtract this from the previous result.
Step 8: Now divide \(253x\) by \(3x\):
\[ \frac{253x}{3x} = \frac{253}{3} \]
Write \(\frac{253}{3}\) above.
4x^2 + 20x + \frac{253}{3}
_________
3x - 13 | 12x^3 + 8x^2 - 7x - 912
- (12x^3 - 52x^2)
_____________________
60x^2 - 7x - 912
- (60x^2 - 260x)
_____________________
253x - 912
- (253x - \frac{3289}{3})
_____________________
-912 + \frac{3289}{3} = \frac{-2736 + 3289}{3} = \frac{553}{3}
Step 9: Write the final expression.
The quotient \(q(x)\) is
\[ q(x) = 4x^2 + 20x + \frac{253}{3} \]
The remainder \(r(x)\) is
\[ r(x) = \frac{553}{3} \]
The divisor \(b(x)\) is
\[ b(x) = 3x - 13 \]
Final Answer:
\[ q(x) + \frac{r(x)}{b(x)} = 4x^2 + 20x + \frac{253}{3} + \frac{\frac{553}{3}}{3x - 13} \]
Thus, the full result can be expressed as:
\[ 4x^2 + 20x + \frac{253}{3} + \frac{553}{3(3x - 13)} \] or
\[ 4x^2 + 20x + \frac{253}{3} + \frac{553}{3(3x - 13)} \] is the answer, expressed as \( q(x) + \frac{r(x)}{b(x)} \) form.