Question

Expressions that occur in calculus are given. Reduce each expression to lowest terms. $$ \frac{(2 x-5) \cdot 3 x^{2}-x^{3} \cdot 2}{(2 x-5)^{2}} $$

Short answer

The simplified expression is \(\frac{4x^3 - 15x^2}{(2x - 5)^2}\)

Step-by-step solution

- Simplify the Numerator

First, expand and simplify the numerator:\(\frac{(2x - 5) \times 3x^2 - x^3 \times 2}{(2x - 5)^2}\)Simplify it to get:\( (2x - 5) \times 3x^2 - 2x^3 = 6x^3 - 15x^2 - 2x^3 = 4x^3 - 15x^2\)

- Simplify the Fraction

Write the simplified form of the expression:\(\frac{4x^3 - 15x^2}{(2x - 5)^2}\)

- Factor Common Terms (If Possible)

Look for common terms in the numerator and the denominator. Notice that there is no common factor that can further simplify the fraction.

- Rewrite the Reduced Form

The expression in its reduced form can't be simplified any further, so the final answer is:\(\frac{4x^3 - 15x^2}{(2x - 5)^2}\)

Key concepts

Rational Expressions

A rational expression is basically a fraction where the numerator and the denominator are both polynomials. Understanding rational expressions is crucial in algebra and calculus.
It's important to know how to simplify rational expressions, which makes solving calculus problems easier.

Factoring Polynomials

Factoring polynomials involves breaking down a polynomial into simpler polynomials that, when multiplied, give the original polynomial.
This concept is essential for simplifying rational expressions and can help identify common factors. Here are key points to remember:

• Always look for a greatest common factor (GCF) first.
• Use techniques like grouping, the difference of squares, or the quadratic formula when applicable.

Simplifying Fractions

Simplifying fractions involves reducing them to their lowest terms. For rational expressions, this process often includes:

• Factoring the numerator and the denominator fully.
• Canceling out common factors that appear in both the numerator and the denominator.

By simplifying fractions, we make complex expressions more manageable and easier to work with in further calculations.