In Exercises 1–16, divide using long division. State the quotient, and the remainder, r(x). (6x³+13x²−11x−15)/(3x²−x−3)

Verified step by step guidance

1. Identify the dividend and divisor: The dividend is $\mathrm{6x^3}+13x^2-11x-15$ and the divisor is $3x^2-x-3$.
2. Divide the leading term of the dividend by the leading term of the divisor: Divide $6x^3$ by $3x^2$ to get the first term of the quotient.
3. Multiply the entire divisor by the term found in step 2 and subtract this product from the dividend to find the new polynomial (the remainder after the first step).
4. Repeat the division process with the new polynomial: divide the leading term of the new polynomial by the leading term of the divisor, multiply the divisor by this term, and subtract again.
5. Continue this process until the degree of the remainder is less than the degree of the divisor. The quotient is the sum of all terms found in each division step, and the remainder is the final polynomial left.

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 in understanding the relationship between these components.

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in the polynomial. In polynomial division, the process continues until the degree of the remainder is less than the degree of the divisor, ensuring the remainder cannot be divided further by the divisor.