Problem 7 - College Algebra

Question

For the following exercises, use long division to divide. Specify the quotient and the remainder. $$ \left(6 x^{2}-25 x-25\right) \div(6 x+5) $$

Step-by-Step Solution

Answer

The quotient is \(x - 5\), and the remainder is \(0\).

Step 1: Set Up the Division

Write the division expression in long division format. Place the dividend, \(6x^2 - 25x - 25\), under the long division symbol and the divisor, \(6x + 5\), outside.

Step 2: Divide the Leading Terms

Divide the leading term of the dividend \(6x^2\) by the leading term of the divisor \(6x\). This calculation gives \(x\), which becomes the first term of the quotient.

Step 3: Multiply and Subtract

Multiply \(x\) by the entire divisor \(6x + 5\) to get \(6x^2 + 5x\). Subtract \(6x^2 + 5x\) from the dividend \(6x^2 - 25x - 25\), which results in \(-30x - 25\).

Step 4: Divide the New Leading Term

Take the new leading term \(-30x\) and divide it by the leading term of the divisor \(6x\). This gives \(-5\), which is the next term of the quotient.

Step 5: Multiply and Subtract Again

Multiply \(-5\) by the divisor \(6x + 5\) to get \(-30x - 25\). Subtract this from the current dividend result \(-30x - 25\), which gives \(0\) as the remainder.

Step 6: State the Quotient and the Remainder

From the above steps, the quotient is \(x - 5\), and the remainder is \(0\). Therefore, the division is exact.

Key Concepts

Understanding the Quotient

The quotient is one of the essential results obtained when performing polynomial long division. It represents the polynomial obtained from dividing two other polynomials. In the exercise provided, when dividing the polynomial \(6x^2 - 25x - 25\) by \(6x + 5\), the quotient is \(x - 5\).

• A quotient can be thought of as the "answer" to a division problem, where the division involves terms divided cleanly without leftover.
• In polynomial division, finding the quotient involves repetitive steps of dividing, multiplying, and subtracting.

To get to the quotient, follow these steps:

• Divide the leading term of the polynomial by the leading term of the divisor.
• Multiply this result by the entire divisor and subtract it from the original polynomial expression, bringing down the next term.
• Repeat this process until you are left with terms whose degree is less than that of the divisor, or you reach a remainder of zero.

In this exercise, the process resulted in a clean division, as shown with no remainder, providing a succinct quotient of \(x - 5\).

Exploring the Remainder

The remainder in polynomial long division is the part of the divided polynomial that cannot be evenly divided by the divisor. In mathematical terms, it is the leftover part once the division process has been executed. In this specific exercise, the remainder is \(0\), indicating a perfect division.

• It provides a crucial check in the division process, confirming whether the division is exact or approximate.
• If the remainder is zero, it tells you that the divisor is a factor of the dividend.

When performing the steps of polynomial division:

• Subtract the product from the current terms so that what is left becomes the next set of terms to divide.
• Continue until the terms left to divide have a degree less than the divisor or until no terms remain, verifying if any remainder is left.

In this exercise, since the remainder is \(0\), \(6x + 5\) is a factor of \(6x^2 - 25x - 25\), meaning \(x - 5\) multiplied by \(6x + 5\) exactly reconstructs the original polynomial.

The Role of Leading Term Division

Leading term division is the first and crucial step in polynomial long division. It involves dividing the leading terms—those with the highest power—of both the dividend and the divisor. This step sets the pace for the rest of the division process.

• In the initial problem, we start by dividing \(6x^2\) by \(6x\), which simplifies to \(x\).
• This forms the beginning of your quotient, guiding the subsequent steps of multiplication and subtraction.

The calculation of the leading term division determines:

• How many times the divisor fits into the dividend's leading terms.
• The first term to appear in the quotient, which is a pivotal element in setting up the rest of the division calculation.

Leading term division gives clarity on how many times to "subtract" your polynomial divisor from the dividend, thus helping to deconstruct the problem into smaller, manageable parts. This step is repeated as new leading terms arrive after partial division results, each contributing to the quotient while verifying exactness when reaching the remainder.