Problem 37 - Elementary Algebra

Question

For the following problems, perform the divisions. $$ \frac{x^{2}+2 x-1}{x+1} $$

Step-by-Step Solution

Answer

Answer: The final result is \(x + 1 - \frac{2}{x + 1}\).

Step 1: Set up the long division

Write the expression as a long division problem, placing the numerator (dividend) under the division symbol and the denominator (divisor) outside. It should look like this: ``` ________ (x+1) | (x^2 + 2x - 1) ```

Step 2: Divide the first terms

Divide the first term of the dividend, \(x^2\), by the first term of the divisor, \(x\). This gives us the first term of the quotient, \(x\). Write this above the corresponding term in the dividend: ``` x ________ (x+1) | (x^2 + 2x - 1) ```

Step 3: Multiply and subtract

Now multiply the first term of the quotient (\(x\)) by the divisor (\(x+1\)) and subtract the result from the dividend: ``` x ________ (x+1) | (x^2 + 2x - 1) -(x^2 + x) _________ x - 1 ```

Step 4: Divide the resulting terms

Now, divide the first term of the resulting polynomial, \(x\), by the first term of the divisor, \(x\). This gives us another term of the quotient, \(1\). Write this term above the corresponding term in the dividend: ``` x + 1 ________ (x+1) | (x^2 + 2x - 1) -(x^2 + x) _________ x - 1 ```

Step 5: Multiply and subtract again

Multiply the new term of the quotient (\(1\)) by the divisor (\(x+1\)) and subtract the result from the remaining polynomial: ``` x + 1 ________ (x+1) | (x^2 + 2x - 1) -(x^2 + x) _________ x - 1 -(x + 1) _________ -2 ```

Step 6: Write the final quotient and remainder

Since the degree of the remaining polynomial (-2) is less than the degree of the divisor (1), the division is complete. The final quotient is \(x + 1\), and the remainder is \(-2\). Write the result as a sum of the quotient and the remainder divided by the divisor: $$ \frac{x^{2}+2 x-1}{x+1} = x + 1 - \frac{2}{x + 1} $$

Key Concepts

Algebra and Polynomial Division

Algebra is the branch of mathematics that deals with symbols and the rules for manipulating those symbols to model and solve real-world problems. One essential algebraic skill is the division of polynomials, a process used to simplify complex expressions or solve equations.

Understanding polynomial division is critical for both basic algebra and more advanced studies in calculus and beyond. The process involves dividing a polynomial, the dividend, by another polynomial, the divisor, to obtain a quotient and possibly a remainder, similar to numerical long division.

For instance, simplifying \( \frac{x^{2}+2x-1}{x+1} \) involves dividing the numerator (x² + 2x - 1) by the denominator (x + 1). The solution reveals both the ratio of the two polynomials and any leftover value, called the remainder. A proper understanding of how to manipulate these expressions allows for effective problem-solving in algebra.

Division of Polynomials

Dividing polynomials may seem daunting at first, but it's simply an extension of the long division you might have learned with numbers. When we talk about polynomial division, we mean the division of two polynomials in which we find a quotient and a remainder.

To divide polynomials, we arrange them similarly to a long division problem, with the dividend inside and the divisor outside. The goal is to determine how many times the divisor can 'fit into' the dividend. Sometimes the division is even, with no remainder, but other times there is a part of the polynomial that cannot be evenly divided, which is then expressed as a remainder.

Using the given example, the division of \(x^{2} + 2x - 1\) by \(x + 1\) yields a quotient of \(x + 1\) with a remainder of -2. This concept allows us to rewrite the original fraction as \(x + 1 - \frac{2}{x + 1}\rm{)}\), expressing the division in a more straightforward form.

Long Division Method

The long division method is a systematic way to divide numbers or expressions, often used when the division is too complex to do mentally. This technique also applies to polynomials, where we deal with variables as well as coefficients.

The process consists of several steps: Dividing the leading terms, multiplying and subtracting, and bringing down subsequent terms until the divisor no longer 'fits' into the remaining polynomial. At its core, it's a method of simplifying and reducing polynomials until what remains cannot be further divided by the divisor.

Illustrative Example:

In our long division example, \(x + 1\) is divided into \(x^{2} + 2x - 1\). We follow the long division algorithm to obtain each term of the quotient until we're left with a remainder of -2, which indicates the end of the process. The methodical nature of long division makes it a reliable tool for students grappling with polynomial division, ensuring that each step is clear and builds upon the previous one.