Problem 10 - Algebra 1

Question

Simplify the expression. $$\frac{x^{2}-4 x+3}{2 x} \div \frac{x-1}{2}$$

Step-by-Step Solution

Answer

The simplified form of the original expression is \( \frac{x - 3}{x} \)

Step 1: Rewrite the Problem

Rewrite the complex fraction expression as a multiplication problem. Rewrite the division as multiplication by taking the reciprocal of the divisor. The expression should then read as follows: \( \frac{x^{2}-4 x+3}{2 x} \times \frac{2}{x-1} \)

Step 2: Simplify the Complex Fraction

Now simplify the smaller fractions before multiplying them. Here, the 2s cancel out, this leaves us with \( \frac{x^{2}-4 x+3}{x} \times \frac{1}{x-1} \)

Step 3: Factor the Polynomial

Factor the polynomial in the numerator. \( x^{2} - 4x + 3 \) can be factored to \( (x-3)(x-1) \). So the expression becomes \( \frac{(x - 3)(x - 1)}{x} \times \frac{1}{x - 1} \)

Step 4: Simplify the Expression

Cancel out the common factors in the numerator and denominator. Here \( x - 1 \) is a common factor, so the expression simplifies to \( \frac{x - 3}{x} \)

Step 5: Check your Result

Ensure that no further simplification is possible in \( \frac{x - 3}{x} \). This is the simplified form of the original expression

Key Concepts

Polynomial Factoring

Polynomial factoring is a key skill in algebra that allows us to simplify expressions and solve equations. It involves expressing a polynomial as the product of its factors. These factors are simpler polynomials that multiply together to give back the original polynomial.
For example, in the expression \( x^2 - 4x + 3 \), you can factor it into \((x-3)(x-1)\). This tells us that if we multiply \((x-3)\) by \((x-1)\), we'll get the original polynomial back. This process:

• Finds simple components of a polynomial.
• Reveals roots of the polynomial equation \( x^2 - 4x + 3 = 0 \).
• Helps simplify algebraic expressions involving the polynomial.

Factoring is particularly useful in simplifying expressions because it allows us to cancel terms, among other things.
This technique is foundational not only in reducing expressions but also in solving quadratic equations and optimizing mathematical solutions.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. They represent mathematical relationships and can be simplified using various algebraic techniques.
In our exercise, the expression \( \frac{x^{2}-4 x+3}{2x} \div \frac{x-1}{2} \) is rearranged and simplified as an algebraic expression.

• Rewriting the division as multiplication: \( \frac{x^{2}-4 x+3}{2x} \times \frac{2}{x-1} \).
• Simplifying involves reducing both numerators and denominators.
• Factoring where possible, as demonstrated in the polynomial \( x^2 - 4x + 3 \).

This transformation from a complex fraction to a simpler fraction is an essential skill in algebra. By making expressions simpler, we make them easier to work with in more complicated equations or when evaluating specific values.

Complex Fractions

Complex fractions are fractions where the numerator, the denominator, or both contain fractions themselves. They can initially look perplexing, but breaking them down step by step makes the process manageable.
The original problem \( \frac{x^{2}-4 x+3}{2x} \div \frac{x-1}{2} \) involves simplifying a complex fraction. This is done by:

• Turning the division into multiplication by the reciprocal: \( \frac{x^{2}-4 x+3}{2x} \times \frac{2}{x-1} \).
• Simplifying step by step: cancelling like terms and reducing where possible.
• Factoring polynomials to make terms cancellable.

The key is to systematically reduce the expression until you reach its most simple form. The final step is to verify no further simplification is possible, resulting in a cleaner and often more intuitive mathematical expression to work with.