In Exercises 1–16, divide using long division. State the quotient, and the remainder, r(x). (6x^3+13x^2−11x−15)/(3x^2−x−3)

Verified step by step guidance

1. Step 1: Set up the long division by writing the dividend $6x^3+13x^2-11x-15$ under the division symbol and the divisor $3x^2-x-3$ outside.
2. Step 2: Divide the first term of the dividend $6x^3$ by the first term of the divisor $3x^2$ to get the first term of the quotient, which is $2x$.
3. Step 3: Multiply the entire divisor $3x^2-x-3$ by $2x$ and subtract the result from the original dividend $6x^3+13x^2-11x-15$.
4. Step 4: Bring down the next term from the dividend to form a new polynomial and repeat the division process with the new polynomial.
5. Step 5: Continue this process until the degree of the remaining polynomial (remainder) is less than the degree of the divisor. The quotient will be the result of the division, and the remaining polynomial will be the remainder.

Key Concepts

Polynomial Long Division

Polynomial long division is a method used to divide polynomials, similar to numerical long division. It involves dividing the leading term of the dividend by the leading term of the divisor, multiplying the entire divisor by this result, and subtracting it from the dividend. This process is repeated with the new polynomial until the degree of the remainder is less than the degree of the divisor.

Quotient and Remainder

In polynomial division, the quotient is the result of the division, representing how many times the divisor fits into the dividend. The remainder is what is left over after the division process, which cannot be divided by the divisor anymore. The relationship 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 polynomial expression. It is crucial in polynomial division because it determines the order of the terms and helps in identifying when to stop the division process. For example, in the polynomial 6x^3 + 13x^2 - 11x - 15, the degree is 3, indicating that the leading term is x^3.