Perform each division. See Examples 9 and 10.
$\frac{(10x^3+11x^2-2x+3)}{(5x+3)}$

Verified step by step guidance

1. Identify the division problem as polynomial long division where you divide the polynomial numerator \$10x^{3} + 11x^{2} - 2x + 3\( by the binomial denominator \)5x + 3$.
2. Set up the long division by writing \$10x^{3} + 11x^{2} - 2x + 3\( under the division bar and \)5x + 3$ outside the division bar.
3. Divide the leading term of the numerator \$10x^{3}\( by the leading term of the denominator \)5x\( to find the first term of the quotient: \)\frac{10x^{3}}{5x} = 2x^{2}$.
4. Multiply the entire divisor \$5x + 3\( by this quotient term \)2x^{2}$, then subtract the result from the original polynomial to find the new remainder.
5. Repeat the process: divide the leading term of the new remainder by \$5x$, multiply the divisor by this new 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 a polynomial by another polynomial of lower degree, 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 remainder has a lower degree 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 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.

Remainder and Quotient in Polynomial Division

When dividing polynomials, the quotient is the result of the division, and the remainder is what is left over when the division cannot continue. The remainder must have a degree less than the divisor, and the original polynomial can be expressed as divisor × quotient + remainder.