Problem 53 - College Algebra

Question

Factor each perfect square trinomial. $$ 4 x^{2}+4 x+1 $$

Step-by-Step Solution

Answer

The factored form of the perfect square trinomial \(4x^2 + 4x + 1\) is \((2x+1)^2\)

Step 1: Identify the Terms

First, identify the terms that make up the trinomial: \(a^2=4x^2\), \(2ab=4x\), and \(b^2=1\). So, a is \(2x\), b is \(1\), and ab stands for \(2x*1\).

Step 2: Check if it is a Perfect Square Trinomial

Determine if the trinomial is a perfect square. To do this, check that \(2ab=4x\), which is indeed the case, thus the trinomial is a perfect square trinomial.

Step 3: Factor the Trinomial

Finally, factor the trinomial into two identical binomials, which yields \((2x+1)(2x+1)\) or \((2x+1)^2\).

Key Concepts

Factoring Trinomials

Factoring trinomials is a fundamental skill in algebra, especially when working with quadratic equations. A trinomial is simply a polynomial with three terms. The key to factoring a trinomial is to identify its structure, which often follows a recognizable pattern.

In the case of perfect square trinomials, like the one in our exercise \( 4x^2 + 4x + 1 \), they factor neatly into a binomial squared. To factor, we look for two identical binomials that multiply to give the original trinomial.

Here's a simple method to factor perfect square trinomials:

• Identify the coefficient of the first term, which is a square, and write it as \( (a)^2 \).
• Check if the middle term fits the form \( 2ab \), where \( a \) and \( b \) are numbers derived from the squared terms.
• Confirm if the last term is a perfect square, like \( (b)^2 \).
• Once identified, write the trinomial as \( (a+b)^2 \).

Understanding this process makes solving such exercises swift and supports mastering more complex algebraic tasks.

Polynomial Identities

Polynomial identities are equations that hold true for every value of the variable involved. They are likened to a set of rules that simplify calculations and help solve polynomial equations. Recognizing a polynomial identity allows us to transform expressions, making them easier to work with.

One familiar polynomial identity is the "perfect square trinomial" identity. It's characterized by the formula: \( (a + b)^2 = a^2 + 2ab + b^2 \). This identity lays the blueprint for what the terms in a perfect square trinomial should look like.

In the exercise \( (2x+1)^2 = 4x^2 + 4x + 1 \), recognizing the identity allows you to see that the trinomial fits perfectly into the pattern specified by the identity.

Being able to identify and manipulate these identities:

• Speeds up the algebraic process.
• Helps in verifying solutions.
• Improves mathematical intuition and fluency.

As you work with various polynomial identities, you'll develop better algebraic reasoning and problem-solving skills.

Algebraic Expressions

Algebraic expressions are the language of algebra, consisting of numbers, variables, and mathematical operations. They serve as a foundational component for expressing mathematical concepts and relationships. Understanding how to work with algebraic expressions is crucial for successfully solving equations and inequalities.

A key aspect of dealing with algebraic expressions is knowing how to manipulate them, such as expanding, simplifying, and factoring. For instance, in our exercise trinomial \( 4x^2 + 4x + 1 \), recognizing it as a perfect square trinomial is part of simplifying it into a manageable form, namely \( (2x + 1)^2 \).

Mastering algebraic expressions involves:

• Understanding the operations being applied.
• Recognizing patterns and common forms, like factoring patterns.
• Being able to decompose expressions into simpler parts.

Through practice and application, solving algebra problems using algebraic expressions becomes intuitive and paves the way for understanding more intricate mathematical concepts.