Problem 31 - Algebra 1: Concepts and Skills

Question

PERFECT SQUARES Factor the expression. $$ 9 x^{2}+6 x+1 $$

Step-by-Step Solution

Answer

The expression \(9x^{2} +6x +1\) can be factored as \((3x+1)^{2}\)

Step 1: Recognize the trinomial

We can see that the given expression is a trinomial. This means it has three terms, and they are \(9x^{2}\), \(6x\), and \(1\). The trinomial is of the form \(a^{2}+2ab+b^{2}\).

Step 2: Identify the values of a and b

For this trinomial to be a perfect square it has to be of the form \((a+b)^2\), which gives \(a^{2}+2ab+b^{2}\). By comparing this with our trinomial, we can identify that \(a\) is \(3x\) and \(b\) is \(1\), since \((3x)^2 = 9x^{2}\), \((2)(3x)(1) = 6x\), and \(1^{2} = 1\).

Step 3: Apply the perfect square formula

We now apply the formula for the square of a binomial, which is \((a+b)^2 = a^{2} + 2ab + b^{2}\). In our case, a is \(3x\) and b is \(1\), we can write \(9x^{2} +6x +1\) as \((3x+1)^{2}\).

Key Concepts

Factoring Trinomials

Trinomials are algebraic expressions with three terms. These can often be rearranged or rewritten in a simpler form through a process called factoring. Factoring trinomials is like untangling the expression so that it can be presented in a more straightforward way.

The trinomial given in our exercise, which is \(9x^2 + 6x + 1\), looks intimidating at first. However, notice it resembles the form \(a^2 + 2ab + b^2\), a clue that it might be a perfect square trinomial. When we recognize this form, our goal is to express the trinomial as the square of a binomial. This helps simplify calculations and understand the expression's behavior, especially when solving equations or inequalities.

Here’s a tip: if a trinomial fits the form \(a^2 + 2ab + b^2\), then it can often be factored as \((a + b)^2\). Identifying values of \(a\) and \(b\) is the key step in this process, allowing us to rewrite the trinomial concisely and verifying that its expanded form matches the original.

Square of a Binomial

The square of a binomial is a fundamental concept in algebra. It refers to an expression that results from multiplying a binomial by itself. The formula for the square of a binomial \((a + b)^2\) can be expanded to \(a^2 + 2ab + b^2\). This pattern helps in recognizing and simplifying expressions in algebra.

In the example trinomial \(9x^2 + 6x + 1\), we notice each term corresponds to the expanded form of a squared binomial. This simplifies to \((3x + 1)^2\). When squared, the expression breaks down to:

• \((3x)^2 = 9x^2\) for the \(a^2\) term
• \(2 \,\cdot\, 3x \,\cdot\, 1 = 6x\) for the \(2ab\) term
• \(1^2 = 1\) for the \(b^2\) term

By recognizing the structure \((a + b)^2\), we can work efficiently through algebraic operations, solving complex problems with less effort and reducing the possibility of errors.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operators such as addition and multiplication. They form the core building blocks of algebra, serving as tools to model real-life situations and abstract mathematical ideas.

Understanding expressions like trinomials and their structures can demystify what looks like a complex arrangement of symbols. The expression \(9x^2 + 6x + 1\) represents more than just its written form. It's a mathematical narrative describing how quantities relate. Breaking it down, as we've done here, offers insights into how different expressions can be manipulated and solved.

By mastering how to factor and simplify algebraic expressions, students gain a greater command over their math skills. This knowledge assists in solving equations, graphing lines, and fully understanding how mathematics models reality. Encouraging curiosity about each part of an expression will lead to deeper comprehension and appreciation of math as a whole.