Problem 50 - Elementary Algebra

Question

For the following problems, perform the multiplications and divisions. $$ (x-5) \div \frac{x-5}{x-2} $$

Step-by-Step Solution

Answer

Answer: The simplified form of the expression is $$x-2$$.

Step 1: Identify the given expression

The given expression is: $$ (x-5) \div \frac{x-5}{x-2} $$ Our task is to simplify this expression by performing multiplication and division.

Step 2: Change the division operation to multiplication by the reciprocal

Instead of dividing by the fraction, we can multiply by the reciprocal of that fraction. The reciprocal of a fraction is obtained by flipping the numerator and denominator. So, the given expression can be written as: $$ (x-5) \times \frac{x-2}{x-5} $$

Step 3: Perform the multiplication

Now we can multiply the expressions by multiplying the numerators with each other and the denominators with each other. $$ \frac{(x-5)(x-2)}{(x-5)} $$

Step 4: Simplify the expression

We can observe that both the numerator and denominator of the fraction have \((x-5)\) as a common factor. So, we can cancel these factors. $$ \frac{(x-5)(x-2)}{(x-5)} = \frac{(x-2)}{(1)} $$ Therefore, the simplified expression is: $$ x-2 $$

Key Concepts

Polynomials

Polynomials are a fundamental concept in algebra. They are expressions that consist of variables and coefficients, involving operations like addition, subtraction, multiplication, and sometimes division. A polynomial is typically expressed in the form: \( a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0 \), where \( n \) is a non-negative integer, and each \( a_i \) is a coefficient. Each term of a polynomial is made up of two parts: a coefficient and a variable raised to a power known as the degree.

• The degree of the polynomial is the highest power of the variable.
• Each variable part with its coefficient is called a term of the polynomial.
• Constant terms are those with no variable part, like \( a_0 \).

Polynomials can be classified based on their degree:

• Linear polynomial: A polynomial with a degree of 1, like \( 3x + 4 \).
• Quadratic polynomial: A polynomial with a degree of 2, like \( 2x^2 + 3x + 5 \).
• Cubic polynomial: A polynomial with a degree of 3, like \( x^3 - 2x^2 + x - 1 \).

Understanding polynomials is crucial because they appear frequently in algebraic simplifications like the one in our expression, where we dealt with \( (x-5) \). By knowing how to manipulate these polynomial expressions, tackling algebra problems becomes much easier.

Fractional Expressions

Fractional expressions are ratios of two polynomials. They are written as fractions where the numerator and denominator are both polynomials. For instance, in the expression \( \frac{x-5}{x-2} \), \( x-5 \) is the numerator, and \( x-2 \) is the denominator.

• Finding a common denominator is often needed when adding or subtracting fractional expressions.
• Simplifying fractional expressions often involves factoring out common terms in the numerator and the denominator.
• Cancelling common factors is a crucial step in the simplification process.

In our worked exercise, simplifying was accomplished by cancelling out \( (x-5) \), showing the importance of identifying common factors. Working with fractional expressions involves similar concepts as with regular fractions, making them easier to manipulate when you remember the basic rules like finding reciprocals or simplifying fractions.

Reciprocal in Mathematics

The reciprocal of a number or an expression is obtained by swapping the numerator and the denominator. For a number \( a \), its reciprocal is \( \frac{1}{a} \). If a fraction is \( \frac{b}{c} \), its reciprocal becomes \( \frac{c}{b} \). This concept is crucial for division of fractions, where division is transformed into multiplication by a reciprocal.

• The reciprocal of a whole number \( n \) is simply \( \frac{1}{n} \).
• The reciprocal of \( \frac{x-5}{x-2} \) would be \( \frac{x-2}{x-5} \).
• Multiplying a number or a fraction by its reciprocal always yields 1, since \( a \times \frac{1}{a} = 1 \).

Understanding the concept of the reciprocal is essential when simplifying complex algebraic expressions, as seen in our step-by-step solution. By replacing division with multiplication by the reciprocal, operations become more straightforward and help in achieving the simplified form of an expression, such as turning \( (x-5) \div \frac{x-5}{x-2} \) into an easier multiplication problem.