Problem 55 - Algebra 2

Question

Factor each expression completely. $$ 12 x^{2}+36 x+27 $$

Step-by-Step Solution

Answer

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

Step 1: Find the Greatest Common Factor (GCF)

First, identify the greatest common factor of the coefficients of the terms. In this case, the GCF of 12, 36, and 27 is 3.

Step 2: Factor out the GCF

Divide each term by the GCF and factor it out. After factoring out 3, the expression becomes 3(4x^2 + 12x + 9).

Step 3: Factor the trinomial

Now, factor the trinomial inside the parentheses, which takes the form \(ax^2 + bx + c\). Look for two numbers that multiply to form \(a \times c\) and add up to \(b\). In this case, the numbers are 3 and 3, so \(4x^2 + 12x + 9\) factors into \(3(2x+3)(2x+3)\) or \(3(2x+3)^2\).

Key Concepts

Greatest Common Factor

Understanding the concept of the greatest common factor (GCF) is crucial when working with algebraic expressions. The GCF is the largest number that divides evenly into all the coefficients (numbers in front of the variables) of the terms in the expression without leaving a remainder. To find the GCF, list out the factors of each coefficient and identify the largest number that appears in every list. For instance, with the coefficients 12, 36, and 27, the common factors are 1 and 3. However, the GCF is the largest of these, which is 3.

In factoring an expression, extracting the GCF is akin to simplifying the problem before addressing more complex aspects. It's like untying a large knot by first loosening the largest loop. Once the GCF is factored out, the remaining expression will often be easier to handle, leading to more efficient and accurate factorization of the trinomial that follows.

Quadratic Expressions

Quadratic expressions are an essential part of algebra and can be recognized by their standard form: \( ax^2 + bx + c \), where \(a\), \(b\), and \(c\) are constants, and \(a\) is not zero. The graph of a quadratic expression is a parabola, which can open upward or downward depending on the sign of \(a\).

When dealing with quadratic expressions, we often need to perform operations like factoring, completing the square, or using the quadratic formula to solve equations. In the context of factoring trinomials, the goal is to break down the quadratic expression into a product of binomials, making it simpler to understand and solve related equations. This process is not only useful for solving equations but also for graphing parabolas and understanding their properties.

Trinomial Factorization

The factorization of trinomials is a process whereby a quadratic expression, which is a polynomial with three terms, is broken down into its simpler binomial factors. This skill is pivotal in solving quadratic equations and simplifying algebraic expressions. The general approach is to look for two binomials whose product recreates the original trinomial.

For a trinomial of the form \( ax^2+bx+c \), the challenge lies in finding two numbers that multiply to give you \( ac \) and, at the same time, add up to \( b \). These numbers are then used to construct the binomials, yielding \((dx+e)(fx+g)\), where \(d \times f = a\), \(e \times g = c\), and \(d \times g + e \times f = b\). This process is akin to a puzzle, where the pieces must fit together perfectly to form the complete picture, which in this case is the original trinomial.