Problem 27 - Elementary and Intermediate Algebra

Question

Factor. $$ 9+4 x^{2}+12 x $$

Step-by-Step Solution

Answer

The expression \(9 + 4x^2 + 12x\) factors to \((2x + 3)^2\).

Step 1: Identify a Common Factor

First, observe if there's a common factor among all the terms in the expression. The expression given is \(9 + 4x^2 + 12x\). There is no common factor across all three terms, so we move to grouping them in a way that reveals common factors.

Step 2: Rearrange and Group Terms

Reorganize the terms to facilitate factoring. The expression can be rearranged to \(4x^2 + 12x + 9\). Now, let's focus on pairing these terms to factor by grouping.

Step 3: Factor by Grouping

Look at the expression \(4x^2 + 12x + 9\). Notice that this can be grouped into \( (4x^2 + 12x) + 9 \). Factor \(4x^2 + 12x\): take out the greatest common factor, \(4x\), leaving you with \(4x(x + 3)\). So now you have \(4x(x+3) + 9\).

Step 4: Recognize a Perfect Square

At a closer look, the expression \(4x(x+3) + 9\) doesn't factor completely by grouping. Therefore, rewrite the expression to check for a perfect square. Notice that rearranging into \((2x + 3)^2\) equals \(4x^2 + 12x + 9\).

Step 5: Verify the Factorization

Verify our factorization \((2x + 3)^2\) by expanding it back: \((2x + 3)(2x + 3)\) gives \(4x^2 + 6x + 6x + 9\), which simplifies to \(4x^2 + 12x + 9\). This confirms that the factoring is correct.

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a special type of polynomial where the expression is the square of a binomial. This concept is key to understanding why certain expressions can be factored into a simpler form.
When dealing with a trinomial like \(4x^2 + 12x + 9\), identifying it as a perfect square can simplify the factoring process. A perfect square trinomial has the form \(a^2 + 2ab + b^2\), which factors into \((a + b)^2\).
In our example, \(4x^2\) is \((2x)^2\) and \(9\) is \(3^2\). The middle term, \(12x\), equals \(2 \cdot 2x \cdot 3\), which confirms the pattern of \(2ab\). Recognizing these components allows us to factor \(4x^2 + 12x + 9\) as \((2x + 3)^2\).

• Understand the basic pattern: Trinomials of form \(a^2 + 2ab + b^2\) are perfect squares and factor to \((a + b)^2\).
• Identify each term: Confirm the square of terms and the middle term fits the \(2ab\) format.

Spotting perfect squares can be a huge time saver in algebra as it leads directly to the factorization without guesswork.

Greatest Common Factor

The concept of the greatest common factor (GCF) is a useful tool in simplifying polynomials. It refers to the largest factor that divides all terms of an expression without leaving a remainder.
In our expression \(9 + 4x^2 + 12x\), the GCF doesn't apply across all terms because there is no single number or variable that divides evenly into 9, 4, and 12. When the GCF approach doesn't directly apply, we can attempt to rearrange and regroup the terms to facilitate factoring.

• Identify potential factors: Look for common numbers or variables in terms.
• Apply when grouping is feasible: Sometimes grouping similar terms or rearranging them can uncover hidden common factors for subsets of the expression.

While not every polynomial benefits from this approach, it's always useful to start by checking for the GCF, as it can simplify calculations and make the factorization process more straightforward.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. They form the building blocks of algebra and embody concepts like polynomials, the focus of our exercise.
Understanding how to manipulate algebraic expressions is crucial in solving equations and simplifying expressions. In particular, factoring involves breaking down expressions into products of simpler factors.
For the expression \(9 + 4x^2 + 12x\), recognizing its structure as a polynomial—a sum of terms with varying powers of \(x\)—is vital. Rearranging terms to remove visual complexity can also unearth opportunities for simplification, such as revealing perfect square trinomials or grouping terms differently for easier factoring.

• Become familiar with terms: Recognize coefficients, variables, and exponents in expressions.
• Master the operations: Learn to perform arithmetic and algebraic operations confidently.

Algebraic expressions require understanding their composition and interactions to perform successful algebraic manipulations.