Problem 34 - Algebra 1: Concepts and Skills

Question

PERFECT SQUARES Factor the expression. $$ 18 x^{2}+12 x+2 $$

Step-by-Step Solution

Answer

The factored form of the expression \(18x^2 + 12x + 2\) is \((3x + 1)^2\).

Step 1: Simplify the expression

Divide all terms in the trinomial \(18x^2 + 12x + 2\) by 2 to simplify the expression. You get \(9x^2 + 6x + 1\).

Step 2: Recognize the pattern

Recognize that this is a perfect square trinomial, that takes the form \(a^2+2ab+b^2\), where a is \(3x\) and b is 1.

Step 3: Factor the trinomial

Using the fact that the factors of a perfect square trinomial \(a^2+2ab+b^2\) are: \((a+b)^2\), you can factor the trinomial into \((3x +1)^2\)

Key Concepts

Perfect Square Trinomials

Perfect square trinomials are special types of algebraic expressions that take the form \(a^2 + 2ab + b^2\). They are called "perfect squares" because these expressions can be factored into the square of a binomial, \((a + b)^2\). This means when you expand \((a + b)^2\), you end up with the perfect square trinomial. Recognizing this pattern can make factoring much simpler.

• To identify a perfect square trinomial, look for three terms: one should be a square, the third term should also be a square, and the middle term should be double the product of their square roots.
• In our example, \(9x^2 + 6x + 1\), notice that \(9x^2\) is a perfect square and \(1\) is a perfect square too. The middle term \(6x\) is twice the product of these two square roots, which confirms the trinomial is a perfect square.

Once identified, you can directly factor it into \((3x + 1)^2\). This quick recognition of pattern helps in factoring easily without more complex calculations.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols like +, -, ×, and ÷. These expressions are the building blocks of algebra, as they allow you to represent real-world problems and equations in a standardized format.

• In the expression \(18x^2 + 12x + 2\), each term is comprised of a coefficient (e.g., 18, 12, and 2) and a variable, \(x\), possibly raised to a power.
• The expression is simplified by dividing each term by 2, leading to \(9x^2 + 6x + 1\), which is a common method to simplify complex expressions and make them easier to work with.

It's crucial to understand how to manipulate these algebraic expressions since it forms the foundation for solving equations and working on higher-level algebra problems.

Polynomial Simplification

Polynomial simplification is the process of transforming a polynomial into its most compact or "simple" form without changing its value. This can involve several steps, such as combining like terms, factoring, or reducing fractions.

• In our example, the original expression is \(18x^2 + 12x + 2\). Initially, it doesn't appear to be a perfect square trinomial, partly due to the coefficients.
• To simplify, divide every term by their greatest common factor, which is 2 in this case, resulting in \(9x^2 + 6x + 1\). This step is crucial for recognizing and using the perfect square trinomial pattern effectively.

After simplification, it often becomes much easier to see factors, patterns, or even further simplify. Understanding this process is key to mastering polynomials and algebra in general.