Problem 69 - Algebra 2

Question

State whether each trinomial is a perfect square. If so, factor it. \(4 x^{2}+12 x+9\)

Step-by-Step Solution

Answer

Yes, it's a perfect square and factors to \((2x + 3)^2\).

Step 1: Identify the Structure of a Perfect Square Trinomial

A perfect square trinomial takes the form \((a+b)^2 = a^2 + 2ab + b^2\). This means that to verify if a given trinomial is a perfect square, it should match the structure \(a^2 + 2ab + b^2\). Check if the given trinomial \(4x^2 + 12x + 9\) follows this pattern.

Step 2: Find Possible Values of 'a' and 'b'

First, identify \(a^2\) and \(b^2\). Here, \(a^2 = 4x^2\) so \(a = 2x\), and \(b^2 = 9\) so \(b = 3\).

Step 3: Verify the Middle Term

The middle term, \(2ab\), must equal \(12x\). Calculate \(2ab = 2 \cdot 2x \cdot 3 = 12x\). Since this matches the given trinomial's middle term, \(4x^2 + 12x + 9\) is a perfect square trinomial.

Step 4: Factor the Trinomial

Since \(4x^2 + 12x + 9\) is confirmed as a perfect square, it factors to \((2x + 3)^2\).

Key Concepts

Factoring Trinomials

Factoring trinomials involves expressing a polynomial of three terms as a product of simpler polynomials. This process aims to simplify expressions and solve equations more easily. When working with trinomials, especially quadratic ones, a common form to encounter is \[ ax^2 + bx + c \]. Where 'a', 'b', and 'c' are constants.

To factor trinomials effectively:

• Identify simple patterns such as perfect square trinomials or difference of squares.
• Understand the structure, like looking for patterns \(a^2 + 2ab + b^2\).
• Use the quadratic formula or factor by grouping if necessary.

Recognizing special forms of trinomials can significantly reduce the complexity of factoring. If a trinomial fits the mold of a perfect square, it implies being expressed as\((a + b)^2\) or \((a - b)^2\), thus making it straightforward to factor. It's vital to practice identifying these forms to become proficient in factoring trinomials.

Algebraic Identities

Algebraic identities are equations that hold true for any values of the variables involved. They are beneficial tools for simplifying complex expressions and verifying solutions in algebra. A fundamental identity relevant to trinomials is the perfect square identity:

• \((a + b)^2 = a^2 + 2ab + b^2\)
• \((a - b)^2 = a^2 - 2ab + b^2\)

In the problem presented, checking if the trinomial is a perfect square involves comparing it to one of these identities.

The given trinomial \(4x^2 + 12x + 9\) can be classified using \((a + b)^2\).
Considering \(a^2 = 4x^2\), \(a\) is \(2x\); for \(b^2 = 9\), \(b\) is 3.We test the middle term, as 2ab should equal 12x, which confirms the identity fits. Mastery of algebraic identities aids students in reorganizing expressions and factoring efficiently.

Polynomial Expressions

Polynomial expressions are sums of terms with variables raised to whole number exponents. Being familiar with them is crucial for solving algebraic equations and performing mathematical operations like addition, subtraction, and multiplication.

A trinomial is a type of polynomial with three terms. Understanding and manipulating these forms is essential, since they appear in various applications and can often reveal more about the solution when deconstructed.Polynomial expressions can be:

• Simplified by combining like terms.
• Factored to expose roots or simplify equations.
• Evaluated at specific values to understand behavior.

Specifically, identifying common patterns like perfect squares in trinomials enables quicker evaluation and understanding. The exercise demonstrated the trinomial \(4x^2 + 12x + 9\). As a perfect square trinomial, simplifying it to \((2x + 3)^2\) showcases the power of recognizing polynomial expressions. This skill is pivotal in both academic settings and practical problem-solving.