In Exercises 1–16, divide using long division. State the quotient, and the remainder, r(x). (6x³+7x²+12x−5)÷(3x−1)

Verified step by step guidance

1. Set up the long division by writing the dividend $\mathrm{6x^3}+\mathrm{7x^2}+\mathrm{12x}-5$ 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: $\backslash frac\{6x^3\}\{3x\}\; =\; 2x^2$.
3. Multiply the entire divisor $3x\; -\; 1$ by the term just found in the quotient $2x^2$, resulting in $2x^2\; \backslash times\; (3x\; -\; 1)\; =\; 6x^3\; -\; 2x^2$. Subtract this from the dividend to find the new remainder.
4. Bring down the next term from the original dividend if necessary, then repeat the division process: divide the leading term of the new remainder by the leading term of the divisor, multiply the divisor by this new quotient term, subtract, and continue.
5. Continue this process until the degree of the remainder is less than the degree of the divisor. The quotient is the collection of all terms found, and the remainder is the final expression left after subtraction.

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 from the dividend, 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, which helps verify the correctness of the division.

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in the polynomial. Understanding the degree is essential in polynomial division because the division process continues until the degree of the remainder is less than the degree of the divisor.