To perform polynomial long division, we need to divide the polynomial \( 12x^3 + 16x^2 + 7x + 112x + 1 \) by the polynomial \( 3x + 13x + 1 \). We can first simplify the divisor \( 3x + 13x + 1 \) to \( 16x + 1 \).
Let's rewrite the polynomial to be divided:
\[ 12x^3 + 16x^2 + 119x + 1 \]
Now, we will divide \( 12x^3 + 16x^2 + 119x + 1 \) by \( 16x + 1 \).
Step 1: Divide the leading terms
- Divide the leading term of the dividend \( 12x^3 \) by the leading term of the divisor \( 16x \): \[ \frac{12x^3}{16x} = \frac{12}{16}x^2 = \frac{3}{4}x^2 \]
Step 2: Multiply and subtract
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Multiply the entire divisor \( 16x + 1 \) by \( \frac{3}{4}x^2 \): \[ \left( \frac{3}{4}x^2 \right)(16x + 1) = 12x^3 + \frac{3}{4}x^2 \]
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Subtract this from the original polynomial: \[ \left( 12x^3 + 16x^2 + 119x + 1 \right) - \left( 12x^3 + \frac{3}{4}x^2 \right) = 16x^2 - \frac{3}{4}x^2 + 119x + 1 \] To subtract the \( x^2 \) coefficients, convert \( 16x^2 \) to a fraction: \[ 16x^2 = \frac{64}{4}x^2 \] So, \[ \frac{64}{4}x^2 - \frac{3}{4}x^2 = \frac{61}{4}x^2 \]
Now we have: \[ \frac{61}{4}x^2 + 119x + 1 \]
Step 3: Repeat the process
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Divide the leading term \( \frac{61}{4}x^2 \) by \( 16x \): \[ \frac{\frac{61}{4}x^2}{16x} = \frac{61}{4 \cdot 16}x = \frac{61}{64}x \]
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Multiply the entire divisor \( 16x + 1 \) by \( \frac{61}{64}x \): \[ \left( \frac{61}{64}x \right)(16x + 1) = \frac{61 \cdot 16}{64}x^2 + \frac{61}{64}x = \frac{61}{4}x^2 + \frac{61}{64}x \]
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Subtract from the current polynomial: \[ \left( \frac{61}{4}x^2 + 119x + 1 \right) - \left( \frac{61}{4}x^2 + \frac{61}{64}x \right) = 119x - \frac{61}{64}x + 1 \]
Convert \( 119x \) to a fraction: \[ 119x = \frac{7616}{64}x \]
Now, subtract: \[ \frac{7616}{64}x - \frac{61}{64}x = \frac{7616 - 61}{64}x = \frac{7555}{64}x \]
So we have: \[ \frac{7555}{64}x + 1 \]
Step 4: Divide again
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Divide \( \frac{7555}{64}x \) by \( 16x \): \[ \frac{\frac{7555}{64}}{16} = \frac{7555}{1024} \]
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Multiply the divisor \( 16x + 1 \) by \( \frac{7555}{1024} \): \[ \frac{7555}{1024}(16x + 1) = \frac{7555 \cdot 16}{1024}x + \frac{7555}{1024} \]
Calculate: \[ \frac{7555 \cdot 16}{1024} = \frac{120880}{1024}x \]
Now we subtract: \[ \left( \frac{7555}{64}x + 1 \right) - \left( \frac{120880}{1024}x + \frac{7555}{1024} \right) \]
Convert \( 1 \) to \( \frac{1024}{1024} \): Resulting in: \[ 1 - \frac{7555}{1024} = \frac{1024 - 7555}{1024} = \frac{-6531}{1024} \]
Final Result
Putting it all together, the quotient of the polynomial division is: \[ Q(x) = \frac{3}{4}x^2 + \frac{61}{64}x + \frac{7555}{1024} \] The remainder is: \[ R(x) = -\frac{6531}{1024} \]
Thus, the division can be expressed as: \[ 12x^3 + 16x^2 + 119x + 1 = (16x + 1)\left(\frac{3}{4}x^2 + \frac{61}{64}x + \frac{7555}{1024}\right) - \frac{6531}{1024} \]