Problem 8 - College Algebra

Question

For the following exercises, use long division to divide. Specify the quotient and the remainder. $$ \left(-x^{2}-1\right) \div(x+1) $$

Step-by-Step Solution

Answer

Quotient: \\(-x + 1\\), Remainder: \\(-2\\).

Step 1: Set Up the Division

Write the dividend \(-x^2 - 1\) and the divisor \(x + 1\) on a long division setup. Place the dividend inside the division bracket and the divisor outside it.

Step 2: Divide the First Term

Determine how many times the first term of the divisor \(x\) goes into the first term of the dividend \(-x^2\). This is \(-x\), as multiplying \(-x\) with \(x\) results in \(-x^2\).

Step 3: Multiply and Subtract

Multiply \(-x\) by the entire divisor \(x + 1\), resulting in \(-x^2 - x\). Subtract this from the dividend \(-x^2 - 1\), which gives \(x - 1\) as the remainder.

Step 4: Bring Down the Next Term

Since there are no other higher degree terms to bring down from the dividend, we don't perform this step.

Step 5: Divide Remaining Term

Consider the new dividend \(x - 1\). Determine how many times \(x\) (the first term of the divisor) goes into the first term of the new dividend \(x\). This is \(1\).

Step 6: Multiply and Subtract Again

Multiply \(1\) by the divisor \(x + 1\) to get \(x + 1\). Subtract \(x + 1\) from \(x - 1\), resulting in \(-2\).

Step 7: Finalize the Solution

The quotient of the division is \(-x + 1\) and the remainder is \(-2\). Thus, the answer is \-x + 1 - \frac{2}{x+1}\.

Key Concepts

Quotient and Remainder

In polynomial long division, two key terms often encountered are the "quotient" and the "remainder". These are crucial in understanding the final result of the division process. To put it simply:

• **Quotient**: This is the result obtained when you divide the dividend by the divisor without considering the remainder (if any).
• **Remainder**: This is what is left over after the division process has been completed. It is the leftover part that cannot be evenly divided by the divisor.

For example, in our solved problem \(rac{-x^2 - 1}{x + 1}\), the quotient is \(-x + 1\) and the remainder is \(-2\). The overall solution can be expressed as:\[-x + 1 - \frac{2}{x + 1}\].
This means, when \(-x^2 - 1\) is divided by \(x + 1\), it goes \(-x + 1\) times fully, leaving \(-2\) as the reminder. Knowing these two terms helps in interpreting the result of any division operation correctly.

Dividend and Divisor

Understanding the terms "dividend" and "divisor" is essential in the polynomial long division process. These terms will show up repeatedly when performing any division operation, which is reflected in our exercise `
` \(-x^2 - 1\) and \(x + 1\). Here's what they mean:

• **Dividend**: This is the polynomial you're dividing; in other words, it's the number inside the long division bracket. In our example, \(-x^2 - 1\) is the dividend. This is what we're breaking down and dividing into smaller parts.
• **Divisor**: This is the polynomial by which you are dividing. It’s placed outside the division bracket. In our example, \(x + 1\) is the divisor. This represents the number of parts you're dividing the dividend into.

Understanding these terms is crucial because they set the stage for the division process and help ensure you're focusing on the correct parts of the expression. Essentially, the job here is to determine how many times the divisor fits into the dividend, which is a key part of solving the division.

Subtract and Multiply

During polynomial division, two major actions you'll perform frequently are subtraction and multiplication. These steps are crucial for breaking down the dividend properly. Here's how it works:

• **Subtract**: Once you've decided what the first part of the quotient should be, you'll multiply it by the entire divisor. Then, subtract this result from the current dividend section. This operation helps reduce the problem down step-by-step. For example, in our problem, we multiplied \(-x\) by \(x + 1\) which gave us \(-x^2 - x\), which we then subtracted from the original dividend part \(-x^2 - 1\), resulting in \(x - 1\).
• **Multiply**: This process involves finding how many times the divisor can go into parts of the dividend. It provides a term of the quotient. After subtracting, the next usage of multiplication is to repeat this cycle until the procedure is complete, refining the quotient with each cycle.

The cycle of subtraction and multiplication continues until every term of the dividend has been accounted for or the terms in the dividend can no longer accommodate the divisor. By performing these operations correctly, you can arrive at the correct quotient and remainder.