Problem 15 - College Algebra and Calculus: An Applied Approach

Question

Factor the perfect square trinomial.\(4 y^{2}+12 y+9\)

Step-by-Step Solution

Answer

The factored form of the perfect square trinomial \(4y^2 + 12y + 9\) is \((2y + 3)^2\).

Step 1: Identify the square roots

First, observe the first and the last term of the trinomial. The square root of \(4y^2\) is \(2y\), and the square root of \(9\) is \(3\).

Step 2: Analyze the middle term

Now, validate if the middle term is twice the product of these identified square roots. Here, \(2 * 2y * 3 = 12y\), which is equal to the middle term of the trinomial.

Step 3: Factor the trinomial

As the trinomial is a perfect-square trinomial, it can be factored into the square of a binomial. Here, the binomial will be the sum of the square roots, which is \((2y + 3)\). The factored form of the trinomial is then \((2y + 3)^2\).

Key Concepts

Factoring Trinomials

Factoring trinomials is a fundamental skill in algebra that involves expressing a trinomial as a product of two binomials. A trinomial is an expression with three terms. For example, in the original exercise, we have the trinomial:

• First Term: \(4y^2\)
• Middle Term: \(12y\)
• Last Term: \(9\)

To factor a trinomial, you need to carefully determine if it fits the pattern of a perfect square trinomial, which allows it to be rewritten as a binomial squared. This pattern helps simplify the process, avoiding lengthy trial and error methods. Instead, you recognize patterns and use them to guide you in writing it more compactly.

Algebra

Algebra is a branch of mathematics dealing with symbols and equations. It's like a mathematical language where placeholders for numbers help solve problems. In trinomials, you seek patterns and relationships among terms.
When factoring, the process involves three key steps:

• Recognize known patterns: like a perfect square trinomial, where the expansion results in familiar term placements.
• Apply mathematical operations: such as multiplication, which breaks down expressions into simpler components.
• Use logical reasoning: to determine the best approach to simplify or solve the expression.

The algebraic approach makes complex expressions manageable, enabling you to manipulate and solve problems efficiently.

Square Roots

Square roots are essential in identifying perfect square trinomials. In the exercise, the focus was on identifying square roots of specific terms:

• For \(4y^2\), the square root is \(2y\), because \((2y)^2 = 4y^2\).
• For \(9\), the square root is \(3\), since \(3^2 = 9\).

Recognizing these square roots helps in determining the components of the binomial created from the trinomial. The process involves verifying that the middle term is two times the product of these square roots. This verification uses multiplication and ensures a trinomial is indeed a perfect square, affirming that it can be simplified further.

Binomial

A binomial consists of two terms. In factoring trinomials, particularly perfect square trinomials, you'll often end up with a binomial squared. The challenge lies in identifying this expression correctly.
For example, from the exercise, the trinomial \(4y^2 + 12y + 9\) factors to the binomial squared:

• \((2y + 3)\)

This binomial captures both the individual components of the original trinomial and the operations needed to achieve its form, namely addition and multiplication. Factoring into a binomial square forms a more manageable expression, facilitating easier manipulation and solution of equations in various algebra problems.