Divide using long division. State the quotient, and the remainder, $r\left(x\right)$. $(6x^3+7x^2+12x-5)÷(3x-1)$

Verified step by step guidance

1. Set up the long division by writing the dividend \(6x^{3} + 7x^{2} + 12x - 5\) under the division bar and the divisor \(3x - 1\) outside the division bar.
2. Divide the leading term of the dividend \$6x^{3}\( by the leading term of the divisor \)3x$ to find the first term of the quotient: \(\frac{6x^{3}}{3x} = 2x^{2}\).
3. Multiply the entire divisor \(3x - 1\) by the term \$2x^{2}$ and subtract the result from the dividend to find the new polynomial to bring down.
4. Repeat the process: divide the new leading term by \$3x$, multiply the divisor by this result, subtract, and continue until the degree of the remainder is less than the degree of the divisor.
5. Express the final answer as the quotient polynomial plus the remainder over the divisor, in the form \(\text{quotient} + \frac{r(x)}{3x - 1}\).

Key Concepts

Polynomial Long Division

Polynomial long division is a method used to divide one polynomial by another, similar to numerical long division. It involves dividing the leading term of the dividend by the leading term of the divisor, multiplying the divisor by this result, subtracting, and repeating until the degree of the remainder is less than the divisor.

Quotient and Remainder in Polynomial Division

When dividing polynomials, the quotient is the polynomial result of the division, and the remainder is the leftover polynomial with a degree less than the divisor. The division can be expressed as Dividend = Divisor × Quotient + Remainder.

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in the expression. Understanding degrees is essential in division because the process continues until the remainder's degree is less than the divisor's degree, indicating the division is complete.