Perform each division. See Examples 9 and 10. (3t²+17t+10)/(3t+2)

Verified step by step guidance

1. Identify the division problem as a polynomial division: divide the polynomial \$3t^2 + 17t + 10\( by the binomial \)3t + 2$.
2. Set up the long division by writing \$3t^2 + 17t + 10\( under the division symbol and \)3t + 2$ outside.
3. Divide the leading term of the dividend \$3t^2\( by the leading term of the divisor \)3t\( to find the first term of the quotient: \)\frac{3t^2}{3t} = t$.
4. Multiply the entire divisor \$3t + 2\( by this term \)t$ and subtract the result from the dividend to find the new remainder.
5. Repeat the process with the new remainder: divide its leading term by \$3t$, 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 terms, multiplying, subtracting, and bringing down the next term until the remainder is of 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 degrees helps determine when to stop the division process, as the division ends when the remainder's degree is less than the divisor's degree.

Remainder and Quotient in Polynomial Division

In polynomial division, the quotient is the result of the division, and the remainder is what is left over when the division cannot continue. The original polynomial equals the divisor times the quotient plus the remainder.