Problem 42 - Algebra 1

Question

Divide. Divide \(4+11 q+6 q^{2}\) by \(2 q+1\)

Step-by-Step Solution

Answer

The quotient after dividing \(4+11q+6q^2\) by \(2q+1\) is \(3q + 2\).

Step 1: Set up the long division

First, arrange the equation so that the divisor \(2q + 1\) is on the outside of the division symbol and the dividend \(4+11q+6q^2\) is under the division symbol. Make sure to write the terms of the dividend in decreasing order of the terms' degrees (degree of a term is the power of variable in the term), so instead of \(4+11q+6q^2\), write it as \(6q^2 + 11q + 4\).

Step 2: Divide the highest degree terms

Divide \(6q^2\), the term with the highest degree in the dividend, by \(2q\), the term with the highest degree in the divisor. The result of this division \(3q\), is the first term of the quotient.

Step 3: Multiply and Subtract

Multiply the divisor \(2q + 1\) by the first term of the quotient \(3q\). Subtract the result from the dividend and bring down the next term. This will provide you a new polynomial.

Step 4: Repeat the Process

Apply the same division process to the new polynomial. Divide the highest degree term of the new polynomial by the highest degree term of the divisor. The result of this will be the next term of the quotient. Continue this process until every term in the dividend has been brought down.

Step 5: Write down the final quotient

All the terms obtained from each division make up the final quotient.

Key Concepts

Long Division

Long division is a method used to divide polynomials, just like we do with numbers. It is especially helpful when dealing with more complex expressions like polynomials. In polynomial long division, you start by setting up the division, placing the divisor outside the division "box" and the dividend inside it.

The main idea is to simplify the division process step by step:

• Divide the first term (highest degree) of the dividend by the first term of the divisor.
• Write the result as a part of your quotient.
• Multiply that part of the quotient by the divisor and subtract it from the dividend.
• Bring down the next term and repeat the process until you have worked through all the terms.

This method ensures a systematic approach to dividing polynomials.

Degree of a Term

The degree of a term in polynomial division is crucial, as it indicates the power of the variable contained within the term. This concept helps us organize the polynomial correctly.

• For instance, in the term \(6q^2\), the degree is 2 because the variable \(q\) is raised to the power of 2.
• In the exercise, we initially rearrange the polynomial \(4 + 11q + 6q^2\) to \(6q^2 + 11q + 4\) based on the degrees: 2 for \(6q^2\), 1 for \(11q\), and 0 for the constant 4.

Ordering by degree makes it easier to manage the long division process step by step.

Divisor and Dividend

Understanding the roles of the divisor and the dividend is essential in polynomial division. The dividend is the polynomial that is being divided, which in this problem is \(6q^2 + 11q + 4\). The divisor is \(2q + 1\), the polynomial by which we divide the dividend.

• The dividend is written inside the division "box," while the divisor sits on the outside.
• We perform operations on these two using long division until we simplify the dividend completely.
• The coefficients of these polynomials interact in the division process to generate the final quotient.

Recognizing how divisors and dividends work together clarifies why we rearrange the polynomial by degrees.

Quotient

The quotient is the result of dividing the dividend by the divisor in polynomial division. Each step of the long division process contributes terms to form the complete quotient.

• For example, in this exercise, we start with dividing \(6q^2\) by \(2q\), resulting in the first term of the quotient: \(3q\).
• As each new term is calculated, it is added to the quotient until all parts of the dividend are addressed.
• The final quotient in this example is the sum of these terms, each resulting from a division step between the highest degree terms left in the dividend and the divisor.

The quotient reflects the completed solution of the division, indicating how many times the divisor fits into the dividend.