Problem 48 - Algebra 1: Concepts and Skills

Question

COMMON FACTOR Factor the expression. $$ 32 x^{2}-48 x+18 $$

Step-by-Step Solution

Answer

The factorized form of the given expression \(32x^{2} - 48x + 18\) is \(2(4x-3)^2\).

Step 1: Identifying the Greatest Common Factor (GCF)

The first step in the process is to find the GCF of the coefficients of \(32x^{2}\), \(48x\), and \(18\). The GCF is the largest value that can divide each of these numbers. Here, it is 2.

Step 2: Factoring out the GCF

Next, divide each term in the expression \(32x^{2}-48x+18\) by the GCF (2), to get: \(16x^{2}-24x+9\).

Step 3: Factoring the resulting quadratic expression

Now, factor the resulting expression \(16x^{2}-24x+9\), which is a perfect square trinomial. A perfect square trinomial can be expressed as \(a^{2}-2ab+b^{2}\) which factors into \((a-b)^{2}\). By comparing the given trinomial with the form \(a^{2}-2ab+b^{2}\), we get \(a = 4x\) and \(b = 3\). Thus, the factorized form is \((4x-3)^2\).

Step 4: Writing the final answer

Combine the GCF from step 2 with the result from step 3. The factorized form of the initial expression is: \(2(4x-3)^2 .\)

Key Concepts

Greatest Common Factor (GCF)

The concept of the Greatest Common Factor (GCF) revolves around finding the largest integer that can evenly divide all coefficients in a polynomial. Consider the polynomial expression given in the exercise:

• The coefficients are 32, 48, and 18.

To find the GCF, identify the largest number that divides each of these coefficients exactly. This process involves:

• Factoring each coefficient into its prime factors:
  • 32 is composed of 2^5,
  • 48 is 2^4 \( \cdot \) 3,
  • 18 breaks down into 2 \( \cdot \) 3^2.
• Determining the smallest power of all common prime factors.
• Multiplying these factors together yields the GCF.

For the given coefficients, the common prime factor is 2, which makes the GCF equal to 2. Utilizing the GCF simplifies the polynomial by dividing it out, making further factoring easier and more manageable.

Perfect Square Trinomials

A perfect square trinomial is a special type of quadratic expression that can be rewritten as the square of a binomial. The standard form of a perfect square trinomial is:

• \(a^2 - 2ab + b^2\) or \(a^2 + 2ab + b^2\)

When factored, it turns into \((a - b)^2\) or \((a + b)^2\) respectively. Identifying a perfect square trinomial involves:

• Checking if both first and last terms are perfect squares.
• Verifying whether the middle term is two times the product of their roots.

In the example, after factoring out the GCF, the expression becomes \(16x^2 - 24x + 9\). Notice:

• \(16x^2\) is a perfect square of \((4x)^2\).
• 9 is a perfect square of 3.
• The middle term \(-24x\) equates to \(-2 \cdot 4x \cdot 3\).

Thus, this trinomial factors perfectly into \((4x - 3)^2\). Recognizing such patterns is crucial in simplifying quadratic expressions that are perfect square trinomials.

Quadratic Expressions

Quadratic expressions are polynomials of degree two, meaning the highest exponent of the variable is two. They generally come in the form:

• \(ax^2 + bx + c\),

where \(a\), \(b\), and \(c\) are constants. Understanding and factoring quadratic expressions is foundational in algebra, as they appear in various mathematical applications, from simple equations to complex modeling.
There are several methods to factor quadratic expressions:

• Using the GCF, which simplifies the polynomial.
• Completing the square method applied for binomials squared.
• Utilizing the quadratic formula for more complex equations.

Factoring quadratic expressions often aims to find values that make the expression equal zero. These are called the roots or solutions of the quadratic equation.
Identifying the perfect square trinomial from the quadratic expression \(16x^2 - 24x + 9\) in the exercise, demonstrates the importance of recognizing patterns and using algebraic identities effectively. Mastering these concepts makes dealing with larger polynomials much more approachable.