In Exercises 1–16, divide using long division. State the quotient, and the remainder, r(x). (2x⁵+9x³+3x²)/(3x²+1)

Verified step by step guidance

1. Identify the dividend and divisor: the dividend is $\mathrm{2x^5}+9\mathrm{x^3}+3\mathrm{x^2}$ and the divisor is $3\mathrm{x^2}+1$.
2. Set up the long division by writing the dividend under the division bar and the divisor outside the bar.
3. Divide the leading term of the dividend $2x^5$ by the leading term of the divisor $3x^2$ to find the first term of the quotient: $\backslash frac\{2x^5\}\{3x^2\}\; =\; \backslash frac\{2\}\{3\}x^3$.
4. Multiply the entire divisor $3x^2\; +\; 1$ by the first term of the quotient $\backslash frac\{2\}\{3\}x^3$, then subtract this product from the dividend to find the new remainder.
5. Repeat the 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 again, and continue until the degree of the remainder is less than the degree of the divisor.

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.

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in the expression. Understanding the degree helps determine when to stop the division process, as the remainder must have a degree less than that of the divisor for the division to be complete.

Quotient and Remainder in Polynomial Division

When dividing polynomials, the quotient is the result of the division, and the remainder is what is left over. The remainder polynomial must have a degree less than the divisor. Expressing the division as dividend = divisor × quotient + remainder is essential for interpreting the result.