Problem 9 - Algebra 1

Question

Simplify the expression. $$\frac{4 x^{2}-25}{4 x} \div(2 x-5)$$

Step-by-Step Solution

Answer

The simplified answer to the problem is \(\frac{x}{2}\).

Step 1: Rewrite the division as the multiplication by the reciprocal

Rewrite the expression \(\frac{4 x^{2}-25}{4 x} \div(2 x-5)\) as \(\frac{4 x^{2}-25}{4 x} * \frac{1}{2 x-5}\)

Step 2: Factor out the numerator and Simplify

Factor out the numerator of the first fraction as \(4(x^2 - 6.25)\). Now the expression becomes: \(\frac{4(x^2 - 6.25)}{4x} * \frac{1}{2x-5}\). Simplify this resulting equation by cancelling out similar terms. By doing this, the equation simplifies to \(\frac{x^2 - 6.25}{x}*\frac{1}{2x-5}\)

Step 3: Simplify further

Rearrange and factor out the shared terms on the equation \(\frac{x^2 - 6.25}{x}*\frac{1}{2x-5}\). This simplifies the equation to: \(x*\frac{1}{2}\). When simplified, the equation becomes: \(\frac{x}{2}\).

Key Concepts

Simplifying Rational Expressions

Simplifying rational expressions is similar to simplifying fractions. It involves reducing a rational expression to its simplest form, making calculations easier.
To simplify a rational expression, start by looking at the numerator and the denominator separately. Check for common factors that can be cancelled out.

• Rewrite any division as multiplication by the reciprocal.
• Factor the expressions fully.
• Cancel out any common factors in the numerator and denominator.

In the original problem, the rational expression \[\frac{4x^2 - 25}{4x} \div (2x-5)\]was rewritten as a multiplication, which is a key first step. By identifying and eliminating the common factors, the expression became much more manageable.

Factoring Polynomials

Factoring polynomials is the process of breaking down a polynomial into the product of its simplest form, or other polynomials.
The strategy often includes:

• Looking for patterns, such as the difference of squares, which is common in problems similar to the one given.
• Identifying any quadratic expressions and using factoring techniques such as splitting the middle term or using formulas.
• Remembering common factoring formulas such as \(a^2 - b^2 = (a+b)(a-b)\).

In our example, the numerator \(4x^2 - 25\) can be recognized as a difference of squares:\[4x^2 - 25 = (2x)^2 - 5^2 = (2x-5)(2x+5)\]By factoring the polynomial, the subsequent simplification becomes more straightforward, ultimately leading to a reduced rational expression.

Algebraic Division

Algebraic division is an important technique when transforming expressions, often seen in polynomial expressions.
Instead of dividing directly, it's useful to multiply by the reciprocal. This helps simplify expressions especially when dealing with polynomials or complex rational expressions.

• Rewrite the division as multiplication by the reciprocal. This changes the operation but not the expression's value.
• Simplify any complex numerators or denominators using factoring or by simplifying each term.
• Ensure you cancel out all common terms correctly to reach the most simplified form.

In the exercise, the division \[\frac{4x^2 - 25}{4x} \div (2x-5)\]was translated into a multiplication by the reciprocal. This change allowed for streamlined factorization and cancellation, simplifying the complex expression to its reduced form \(\frac{x}{2}\). These steps showcase the elegant simplicity achievable through fundamental algebraic principles.