Problem 36 - Algebra for College Students

Question

Perform the indicated divisions. $$ \frac{3 x^{3}+2 x^{2}-5 x-1}{x^{2}+2 x} $$

Step-by-Step Solution

Answer

Quotient is \(3x - 4\) with a remainder \(3x - 1\).

Step 1: Set Up the Long Division

Start by arranging the problem in a long division format, where the dividend is \(3x^3 + 2x^2 - 5x - 1\) and the divisor is \(x^2 + 2x\). Write these in a long division bar setup.

Step 2: Divide the Leading Terms

Divide the leading term of the dividend, \(3x^3\), by the leading term of the divisor, \(x^2\). This gives \(3x\). Write \(3x\) above the division bar.

Step 3: Multiply and Subtract

Multiply \(3x\) by the entire divisor \(x^2 + 2x\), resulting in \(3x^3 + 6x^2\). Subtract this product from the current dividend \(3x^3 + 2x^2 - 5x - 1\), leaving \(-4x^2 - 5x - 1\).

Step 4: Repeat Division with New Dividend

Now, divide the new leading term \(-4x^2\) by \(x^2\), which equals \(-4\). Write \(-4\) above the division bar, next to \(3x\).

Step 5: Multiply and Subtract Again

Multiply \(-4\) by the entire divisor \(x^2 + 2x\), resulting in \(-4x^2 - 8x\). Subtract this from \(-4x^2 - 5x - 1\), leaving \(3x - 1\).

Step 6: Identify the Remainder

Since the degree of the remaining polynomial \(3x - 1\) is less than the degree of the divisor \(x^2 + 2x\), \(3x - 1\) is the remainder.

Step 7: Write the Final Answer

The final answer is the quotient plus the remainder over the divisor: \(3x - 4 + \frac{3x - 1}{x^2 + 2x}\).

Key Concepts

Long Division in Algebra

In algebra, long division is a method used to divide polynomials, much like the long division method used with numbers. To perform long division, you align the terms of the polynomial expression, placing the dividend under the long division bar and the divisor outside it. The process involves dividing the leading term of the dividend by the leading term of the divisor. This gives you the first term of the quotient.

Here is a simple approach:

• Align terms according to their degrees.
• Divide the leading term of the dividend by the leading term of the divisor.
• Multiply the divisor by the result and subtract from the dividend.
• Repeat with the new polynomial formed until the degree of the new polynomial is less than the degree of the divisor.

This method helps break down complex polynomials into simpler parts, making it easier to understand their structure.

Degree of Polynomials

The degree of a polynomial is a key concept in polynomial division. It refers to the highest power of the variable in the polynomial. Understanding the degree is crucial because:

• It helps in organizing terms in descending order, which is essential for long division.
• The degree informs us when to stop the division process—specifically, when the degree of the remainder is less than the degree of the divisor, the division concludes.

The degree plays a crucial role in polynomial division, ensuring proper alignment and effective operation during long division. For example, in the given polynomial division problem, the divisor's degree is 2, and you'll stop dividing when the remainder's degree is less than 2.

Remainder Theorem

The remainder theorem is a handy tool in algebra that simplifies the process of finding the remainder of a polynomial division. It states that for a polynomial \( f(x) \), when divided by \( x - c \), the remainder is \( f(c) \). For polynomial division involving higher-degree divisors, the remainder theorem helps validate results and understand the relation between divisors and remainders.

While we directly perform long division for polynomial division, it's helpful to know that for straightforward divisions, this theorem offers an alternative check by substituting the root of the divisor into the dividend polynomial to find the remainder.

Rational Expressions

Rational expressions are fractions where the numerator and the denominator are polynomials. It's crucial to comprehend these expressions as they appear frequently in algebraic equations. When working with rational expressions, simplifying the fraction by dividing polynomials is an essential step.

For example, the division of two polynomials results in a quotient and a remainder, which can be rewritten as a mixed rational expression. In this context, understanding how to perform polynomial long division allows for the simplification of complex rational expressions, providing both insights into their structure and practical solutions.