Problem 19 - Elementary and Intermediate Algebra

Question

Determine whether each of the following is a perfect-square trinomial. $$ 4 y^{2}-12 y+9 $$

Step-by-Step Solution

Answer

The trinomial \(4y^2 - 12y + 9\) is a perfect-square trinomial.

Step 1: Identify the square terms

A perfect-square trinomial takes the form \(a^2 - 2ab + b^2\). First, identify if the first and last terms are perfect squares. The first term, \(4y^2\), is \((2y)^2\) and the last term, \(9\), is \(3^2\).

Step 2: Check the middle term

For the trinomial to be a perfect-square trinomial, the middle term must be equal to \(-2ab\). We have that \(a = 2y\) and \(b = 3\), which gives us \(-2ab = -2(2y)(3) = -12y\).

Step 3: Compare with given trinomial

Verify if the middle term from step 2, \(-12y\), matches the middle term of the given trinomial, \(-12y\). Since they are equal, this confirms the trinomial is a perfect-square trinomial.

Step 4: Conclude with the trinomial structure

Since all conditions for a perfect-square trinomial are satisfied, we can conclude that \(4y^2 - 12y + 9 = (2y - 3)^2\).

Key Concepts

Understanding Algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating these symbols. The symbols in algebra, often letters like \(x\) or \(y\), can represent numbers and quantities in various scenarios. Equations and formulas in algebra are used to express relationships between these symbols.

When dealing with polynomials, algebra allows us to explore and simplify expressions. For example, a trinomial is a specific type of polynomial with three terms. Understanding how each part of an algebraic expression relates to others is key. In trinomials, recognizing patterns like perfect-square trinomials can make complex problems simpler. This involves using established formulas to easily factor and manipulate the expressions.

Trinomial Factorization Techniques

Trinomial factorization is the process of breaking down a trinomial into simpler binomial factors. One special type of trinomial factorization involves recognizing a perfect-square trinomial. Perfect-square trinomials fit the pattern:

• \( a^2 - 2ab + b^2 \)

For example, consider \(4y^2 - 12y + 9\).

This expression can be factored using the formula above. First, confirm that the first and last terms are perfect squares:

• The first term, \(4y^2\), is \((2y)^2\).
• The last term, 9, is \(3^2\).

Next, check the middle term.

• Use \(-2ab\), where \(a = 2y\) and \(b = 3\).
• This gives \(-2(2y)(3) = -12y\), which matches our middle term.

Since all conditions of a perfect-square trinomial are satisfied, the trinomial is factored into \((2y - 3)^2\). This method simplifies handling trinomials and is a helpful technique of algebra.

Exploring Polynomials

Polynomials are expressions consisting of variables and coefficients, composed through operations of addition, subtraction, and multiplication. Each polynomial is categorized by its degree, the highest power of its variable. Trinomials are a class of polynomials, featuring three terms combined in the form \(ax^2 + bx + c\).

Understanding the structure of polynomials is crucial for simplification and solving algebraic equations. A special case of polynomials is the perfect-square trinomial, as seen in \(4y^2 - 12y + 9\). Recognizing this allows us to rewrite the expression as \((2y - 3)^2\), showcasing the power and efficiency of algebraic manipulation.

By recognizing how the perfect square trinomial formula works, students can factor or solve polynomial expressions more effectively, establishing a deeper understanding of algebraic principles.