Problem 54 - Elementary and Intermediate Algebra

Question

Simplify. See Example \(5 .\) $$ \frac{9 y^{2}-12 y+4}{(3 y-2)^{3}} $$

Step-by-Step Solution

Answer

\(\frac{1}{3y - 2}\)

Step 1: Identify the Numerator

The numerator of the expression is given as \(9y^{2} - 12y + 4\). Recognize it as a quadratic expression which might be factored further.

Step 2: Factor the Numerator

Notice that \(9y^2 - 12y + 4\) is a perfect square trinomial. Try factoring it in the form of \((ay - b)^2\). Observing closely, we find: \((3y - 2)^2\).

Step 3: Write the Denominator

The denominator is given as \((3y - 2)^3\). Keep this expression unchanged for now, as we will use it to simplify the fraction.

Step 4: Simplify the Fraction

The fraction is \(\frac{(3y - 2)^2}{(3y - 2)^3}\). Now, simplify by canceling out common terms: \(\frac{1}{3y - 2}\).

Step 5: Write the Final Simplified Expression

The simplified expression after canceling the common terms from the numerator and denominator is \(\frac{1}{3y - 2}\).

Key Concepts

Factoring Quadratics

Factoring quadratics is an essential skill in algebra that helps simplify expressions and solve equations. A quadratic expression typically takes the form ax^2 + bx + c, where a, b, and c are constants, and x is the variable. The goal is to rewrite this expression as a product of two binomials. To factor a quadratic:

• Firstly, determine if the quadratic can be simplified into a form like (px + q)(rx + s).
• Check if it is a perfect square trinomial, more on this later.
• Use techniques like factoring by grouping or trial and error to find two numbers that multiply to give ac and add up to b.

Factoring is not only used for simplification but also in understanding the properties of quadratic equations. Practice by identifying patterns and using them to factor successfully.

Perfect Square Trinomials

A perfect square trinomial is a special type of quadratic where the first and third terms are perfect squares, and the middle term is twice the product of their square roots. An expression like this takes the form: a^2 + 2ab + b^2, and can be rewritten as (a+b)^2. Let's explore how to identify and factor these lovely formations:

• Recognize expressions similar to x^2 + 2xy + y^2 or x^2 - 2xy + y^2.
• The middle term should be exactly twice the product of the square roots of the first and last terms.
• Simplify the expression to (a ± b)^2 depending on the sign of the middle term.

This method significantly simplifies your work, especially when simplifying rational expressions, as it reduces the complexity of the expression quickly.

Algebraic Fractions

Algebraic fractions, similar to numerical fractions, involve variables in their numerators and/or denominators. Simplifying these expressions often requires an understanding of factoring, common terms, and cancellation. Here's how to navigate through algebraic fractions:

• Check if both the numerator and the denominator can be factored.
• Cancel out any common factors between the numerator and denominator.
• Always express your final result in the simplest form, ensuring the expression is as compact as possible.

It is crucial to master these steps as they not only simplify expressions but also aid in solving more complex equations quickly. Simplifying algebraic fractions helps in reducing computational complexity when dealing with polynomial expressions.