Question

introduction homework (standard 0)
score: 4/28 answered: 4/25
question 5
find the quotient and remainder using long division.
\\(\frac{12x^3 - 8x^2 - 29x + 16}{3x - 5}\\)
the quotient is
the remainder is
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Explanation:

Step1: Divide the leading terms

Divide the leading term of the dividend \(12x^3\) by the leading term of the divisor \(3x\). We get \(\frac{12x^3}{3x} = 4x^2\). This is the first term of the quotient.
Multiply the divisor \(3x - 5\) by \(4x^2\): \(4x^2(3x - 5)=12x^3 - 20x^2\).
Subtract this from the dividend: \((12x^3 - 8x^2 - 29x + 16)-(12x^3 - 20x^2)=12x^2 - 29x + 16\).

Step2: Divide the new leading terms

Now, divide the leading term of the new dividend \(12x^2\) by the leading term of the divisor \(3x\). We get \(\frac{12x^2}{3x}=4x\). This is the next term of the quotient.
Multiply the divisor \(3x - 5\) by \(4x\): \(4x(3x - 5)=12x^2 - 20x\).
Subtract this from the new dividend: \((12x^2 - 29x + 16)-(12x^2 - 20x)= - 9x + 16\).

Step3: Divide the new leading terms again

Divide the leading term of the new dividend \(-9x\) by the leading term of the divisor \(3x\). We get \(\frac{-9x}{3x}=-3\). This is the next term of the quotient.
Multiply the divisor \(3x - 5\) by \(-3\): \(-3(3x - 5)=-9x + 15\).
Subtract this from the new dividend: \((-9x + 16)-(-9x + 15)=1\).

Answer:

The quotient is \(4x^2 + 4x - 3\)
The remainder is \(1\)