Problem 44 - Algebra for College Students

Question

Perform the indicated divisions. $$ \left(4 a^{3}-2 a^{2}+7 a-1\right) \div\left(a^{2}-2 a+3\right) $$

Step-by-Step Solution

Answer

The quotient is \(4a + 6\) and the remainder is \(7a - 19\).

Step 1: Set up the Polynomial Division

To divide polynomials, we'll use long division. Write the dividend \(4a^3 - 2a^2 + 7a - 1\) under the division symbol and the divisor \(a^2 - 2a + 3\) outside the division symbol.

Step 2: Divide the Leading Terms

Divide the leading term of the dividend \(4a^3\) by the leading term of the divisor \(a^2\), which gives \(4a\). Write \(4a\) as the first term of the quotient above the division line.

Step 3: Multiply and Subtract

Multiply the entire divisor \(a^2 - 2a + 3\) by the term \(4a\). The result is \(4a^3 - 8a^2 + 12a\). Subtract this from the dividend \(4a^3 - 2a^2 + 7a - 1\), resulting in the new polynomial \(6a^2 - 5a - 1\).

Step 4: Repeat the Division Process

Divide the new leading term \(6a^2\) by the leading term of the divisor \(a^2\), which gives \(6\). Write \(6\) as the next term of the quotient.

Step 5: Multiply and Subtract Again

Multiply the divisor \(a^2 - 2a + 3\) by \(6\) to get \(6a^2 - 12a + 18\). Subtract \(6a^2 - 12a + 18\) from the current dividend \(6a^2 - 5a - 1\), resulting in \(7a - 19\).

Step 6: Determine the Remainder

Since the degree of the remainder \(7a - 19\) is less than the degree of the divisor \(a^2 - 2a + 3\), the division process stops here. The remainder does not continue division.

Key Concepts

Long Division

Long division is a method used to divide polynomials, similar to how it’s used with numbers. When you perform polynomial long division, you systematically find out how many times the divisor fits into parts of the dividend.
Like numerical long division, the goal is to take a large problem and break it into smaller, more manageable pieces. Here’s how the process works:

• First, you write the dividend inside the division symbol and the divisor outside.
• Next, determine how many times the leading term of the divisor fits into the leading term of the dividend.
• This term then becomes a part of your quotient.
• After that, you multiply the entire divisor by this quotient term and subtract to find the new dividend.
• The process repeats until the remainder's degree is less than the divisor's degree.

This structured approach makes long division a reliable way to divide polynomials and gain insight into polynomial relationships.

Divisor

In polynomial division, the divisor is the polynomial that you are dividing the original polynomial by. For our problem, the divisor is the polynomial \(a^2 - 2a + 3\). Understanding the structure of the divisor is key to solving the division problem.
During the process, we focus on the leading term of the divisor to derive each step of the division. This leading term, in this case \(a^2\), helps in determining how many times it fits into any leading term of the dividend. The aim is to eliminate the leading term of the dividend by subtracting an appropriate multiple of the divisor.
The divisor stays constant throughout the division process, which is why careful attention must be paid to it early on to ensure accurate calculations later in the procedure.

Dividend

The dividend in polynomial division is the polynomial that you're dividing. It's placed under the division symbol and divided by the divisor step by step. In our exercise, the dividend is \(4a^3 - 2a^2 + 7a - 1\).
Each stage of the division focuses on reducing this dividend by removing multiples of the divisor, ultimately simplifying as much as possible. This process involves dividing the dividend's leading term by the divisor's leading term to generate terms of the quotient.
The remaining polynomial after subtracting the product of this quotient term and the divisor becomes the new dividend for the next step, allowing the division to continue.

Remainder

In polynomial division, the remainder is what is left over once the division process is complete. It happens when the degree of the remaining polynomial is less than the degree of the divisor.
For the problem at hand, the remainder is \(7a - 19\). This indicates the division stopped when further factoring wasn’t possible beyond this point. The remainder becomes an important indicator in polynomial division because it essentially tells you how close you got to dividing evenly.
The remainder also can be expressed as a fraction of the original divisor when recounting the complete division, usually by stating the quotient plus the remainder over the divisor.