Questions: Divide using long division. State the quotient, q(x), and the remainder, r(x). (12 x^2 -7 x -3) ÷ (3 x -4) (12 x^2 -7 x -3) ÷ (3 x -4) =

Solution Steps

To solve the problem of dividing the polynomial \(12x^2 - 7x - 3\) by \(3x - 4\) using long division, follow these steps:

1. Divide the leading term of the dividend \(12x^2\) by the leading term of the divisor \(3x\) to get the first term of the quotient.
2. Multiply the entire divisor by this term and subtract the result from the original polynomial.
3. Repeat the process with the new polynomial obtained after subtraction until the degree of the remainder is less than the degree of the divisor.
4. The result will be the quotient and the remainder.

We are tasked with dividing the polynomial \(12x^2 - 7x - 3\) by \(3x - 4\). This can be expressed as: \[ \frac{12x^2 - 7x - 3}{3x - 4} \]

1. Divide the leading term of the dividend \(12x^2\) by the leading term of the divisor \(3x\): \[ \frac{12x^2}{3x} = 4x \]
2. Multiply the entire divisor \(3x - 4\) by \(4x\): \[ 4x(3x - 4) = 12x^2 - 16x \]
3. Subtract this result from the original polynomial: \[ (12x^2 - 7x - 3) - (12x^2 - 16x) = 9x - 3 \]

Now, we repeat the process with the new polynomial \(9x - 3\):

1. Divide the leading term \(9x\) by the leading term \(3x\): \[ \frac{9x}{3x} = 3 \]
2. Multiply the entire divisor \(3x - 4\) by \(3\): \[ 3(3x - 4) = 9x - 12 \]
3. Subtract this from \(9x - 3\): \[ (9x - 3) - (9x - 12) = 9 \]

Final Answer