Problem 63 - Intermediate Algebra

Question

Perform each division. Divide \(2 s^{2}+13 s+5\) by \(2 s+3\)

Step-by-Step Solution

Answer

The quotient is \(s + 5\) with a remainder of \(-10\).

Step 1: Set up the Division

We will perform polynomial long division, similar to numerical long division. Start by setting up the expression \((2s^2 + 13s + 5) \div (2s + 3)\). Write it in long division format with \(2s + 3\) as the divisor and \(2s^2 + 13s + 5\) as the dividend.

Step 2: Divide the Leading Terms

Compare the leading term of the dividend \(2s^2\) with the leading term of the divisor \(2s\). Divide \(2s^2\) by \(2s\) to get the first term of the quotient, which is \(s\). Write \(s\) above the division line.

Step 3: Multiply and Subtract

Multiply the entire divisor \(2s + 3\) by the term obtained in Step 2, which is \(s\). This gives \(2s^2 + 3s\). Subtract \(2s^2 + 3s\) from \(2s^2 + 13s + 5\) to get a new dividend: \(10s + 5\).

Step 4: Repeat Division for the New Dividend

Divide the leading term of the new dividend \(10s\) by the leading term of the divisor \(2s\) to get \(5\). Write \(+5\) in the quotient.

Step 5: Multiply and Subtract Again

Multiply \(2s + 3\) by \(5\), obtaining \(10s + 15\). Subtract \(10s + 15\) from \(10s + 5\) to get \(-10\), which is the remainder.

Step 6: Write the Complete Answer

Combine the quotient obtained in Steps 2 and 4 with the remainder. The answer is \(s + 5\) with a remainder of \(-10\), which can be written as \(s + 5 + \frac{-10}{2s+3}\).

Key Concepts

Leading Term

The leading term in polynomial long division is the first term of a polynomial when it is expressed in standard form (highest degree first). This term plays a crucial role in the division process. In the division of \[2s^2 + 13s + 5 \div 2s + 3,\] the leading terms are the parts with the highest powers of the variable 's'. For the dividend \(2s^2 + 13s + 5\), the leading term is \(2s^2\), while for the divisor \(2s + 3\), it is \(2s\). Understanding and identifying the leading term is essential because it is used to determine how many times the divisor can "fit into" the dividend at each step.

• The division process begins with dividing the leading term of the dividend by the leading term of the divisor,
• In our example, this division gives the first term of the quotient \(s\).

This approach helps simplify the polynomial and guide you toward the solution.

Division

Polynomial division is similar to normal long division. You aim to divide the dividend by the divisor step-by-step. Start by arranging the division expression in the **long division format**. Next, divide the leading term of the dividend by the leading term of the divisor. Once you have the first term of your quotient, use it to multiply the entire divisor. Subtract this result from the original or remaining dividend.

• This subtraction gives a new polynomial (new dividend).
• You repeat the division process with this new polynomial.

The overall objective is to continue this process until the degree of the new dividend is less than that of the divisor.

Remainder

In polynomial long division, the remainder is what’s left over after you have divided as much as possible using the divisor. In the example given, after subtracting\[10s + 15\] from\[10s + 5,\] we arrive at a remainder of \(-10\). The remainder is part of the final answer and is often expressed as a fraction added to the quotient. Since the degree of the remainder is less than the degree of the divisor, division stops here.

• This makes the remainder \[\frac{-10}{2s+3}.\]

Writing the remainder correctly is crucial as it completes the division problem's solution.

Quotient

The quotient in polynomial long division is the result obtained from dividing the dividend by the divisor, excluding the remainder. In the division of \[2s^2 + 13s + 5 \div 2s + 3,\]the quotient evolves during the division process. Initially, dividing the leading terms gives the first term \(s\). Repeating this division with the new dividend yields \(+5\), therefore making the entire quotient \[s + 5.\]The quotient represents how many times the divisor fits into the dividend or its parts.

• The quotient, combined with the remainder, gives you the complete answer:
• \(s + 5 + \frac{-10}{2s+3}\)

Understanding how the quotient is built up step-by-step is vital for fully grasping polynomial long division.