Problem 67 - Algebra for College Students

Question

Recall that the indicated quotient of a polynomial and its opposite is \(-1\). For example, \(\frac{x-2}{2-x}\) simplifies to \(-1\). Keep this idea in mind as you add or subtract the following rational expressions. (a) \(\frac{1}{x-1}-\frac{x}{x-1}\) (b) \(\frac{3}{2 x-3}-\frac{2 x}{2 x-3}\) (c) \(\frac{4}{x-4}-\frac{x}{x-4}+1\) (d) \(-1+\frac{2}{x-2}-\frac{x}{x-2}\)

Step-by-Step Solution

Answer

(a) -1, (b) -1, (c) 0, (d) -2

Step 1: Simplify Expression (a)

For expression (a), \( \frac{1}{x-1} - \frac{x}{x-1} \), the denominators are the same. Combine the numerators: \( 1 - x \). So, \( \frac{1}{x-1} - \frac{x}{x-1} = \frac{1-x}{x-1} \). Since \( x-1 \) and \( 1-x \) are opposites, by using the quotient property, we have \( \frac{1-x}{x-1} = -1 \).Therefore, the result is \(-1\).

Step 2: Simplify Expression (b)

For expression (b), \( \frac{3}{2x-3} - \frac{2x}{2x-3} \), the denominators are the same. Combine the numerators: \( 3 - 2x \). So, \( \frac{3}{2x-3} - \frac{2x}{2x-3} = \frac{3-2x}{2x-3} \).Since \( 2x-3 \) and \( 3-2x \) are opposites, we use the quotient property, giving \( \frac{3-2x}{2x-3} = -1 \).Thus, the result is \(-1\).

Step 3: Simplify Expression (c)

For expression (c), \( \frac{4}{x-4} - \frac{x}{x-4} + 1 \), start by simplifying \( \frac{4-x}{x-4} \) (combine the first two fractions): \( \frac{4-x}{x-4} = -1 \) because \( 4-x \) and \( x-4 \) are opposites.Adding \(-1\) to \(1\) gives: \(-1 + 1 = 0\).Thus, the result is \(0\).

Step 4: Simplify Expression (d)

For expression (d), \(-1 + \frac{2}{x-2} - \frac{x}{x-2} \), simplify \( \frac{2-x}{x-2} \) (the second term):\( \frac{2-x}{x-2} = -1 \) because \( 2-x \) and \( x-2 \) are opposites.Combine with -1: \(-1 - 1 = -2\).So, the final result is \(-2\).

Key Concepts

Polynomial Quotient

Polynomials are mathematical expressions involving variables and coefficients. Generally, polynomials take the form: \[ ax^n + bx^{n-1} + ext{...} + k \] where \( a, b, \) and \( k \) are constants. The process of dividing one polynomial by another is called finding the polynomial quotient. When the two polynomials are opposites, the quotient results in \(-1\). For example, in the expression \( \frac{x-2}{2-x} \), the numerator and denominator are opposites. A neat property of polynomials is that dividing by an opposite always simplifies to \(-1\). This can be useful in simplifying more complex algebraic expressions.

• Tip 1: Always look for opposite terms, especially useful in subtraction problems.
• Tip 2: Remember that flipping the subtraction sign can change the whole expression.

Take your time to familiarize yourself with recognizing polynomial opposites, as it's a powerful tool in simplifying terms.
Understanding these basics will make working with polynomial quotient problems much easier.

Simplifying Expressions

Simplifying expressions refers to reducing expressions into their most basic and understandable form. In algebra, this often involves combining like terms or factoring expressions to make them simpler. The main goal is to make an expression as clean and straightforward as possible, which often simplifies subsequent calculations.
When looking at rational expressions where the denominators are identical, such as \( \frac{1}{x-1} - \frac{x}{x-1} \), it is viable to simply combine the numerators over the common denominator. This is a primary tactic used to simplify expressions quickly. When working through expressions, always:

• Check if terms can be combined.
• Identify and factor common factors.
• Recognize patterns that may help reduce complexity.

These strategies are key in transforming complicated expressions into easily manageable forms, and are essential skills for solving algebra problems efficiently.

Algebraic Opposites

In algebra, recognizing opposites can simplify expressions drastically. Algebraic opposites are pairs of expressions that add up to zero when combined. For instance, \( x-1 \) and \( 1-x \) are opposites because combining them results in zero. The principle of opposites is handy because dividing a polynomial by its opposite always equals \(-1\).
When adjusting expressions using opposites, it's useful to remember:

• Opposites are a specific set of expressions; keep an eye out for signs switch.
• Combining terms to create opposites can simplify operations significantly.
• This knowledge works well with subtraction operations.

Using algebraic opposites can turn a challenging math problem into a simpler one by reducing fractions to a basic level, as seen in the exercise given. Recognizing and leveraging opposites is a fundamental algebra skill.