Question

1. \frac{12x^4 - 4x^3 + 13x^2 + 2x + 1}{3x^2 - x + 4}

Explanation:

Step1: Perform Polynomial Long Division

Divide the leading term of the numerator \(12x^4\) by the leading term of the denominator \(3x^2\) to get the first term of the quotient: \(\frac{12x^4}{3x^2} = 4x^2\). Multiply the entire denominator \(3x^2 - x + 4\) by \(4x^2\): \(4x^2(3x^2 - x + 4)=12x^4 - 4x^3 + 16x^2\). Subtract this from the numerator: \((12x^4 - 4x^3 + 13x^2 + 2x + 1)-(12x^4 - 4x^3 + 16x^2)= - 3x^2 + 2x + 1\).

Step2: Continue the Division

Now, divide the leading term of the new numerator \(-3x^2\) by the leading term of the denominator \(3x^2\) to get \(-1\). Multiply the denominator \(3x^2 - x + 4\) by \(-1\): \(-1(3x^2 - x + 4)= - 3x^2 + x - 4\). Subtract this from the current numerator: \((-3x^2 + 2x + 1)-(-3x^2 + x - 4)=x + 5\).

Answer:

The result of the division is \(4x^2 - 1+\frac{x + 5}{3x^2 - x + 4}\)