Problem 40 - Algebra 1: Concepts and Skills

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify the common factor

Firstly, check whether there's a common factor for both terms \(-32\) and \(18 x^{2}\). Indeed, both terms are divisible by 2, thus 2 is a common factor.

Step 2: Factor out the common factor

The common factor identified is 2. Therefore, factor 2 out of the expression to obtain \(2(-16 + 9x^{2})\).

Step 3: Rearrange the terms

We can rearrange the terms inside the parenthesis for the expression to follow the standard format with square terms coming first. Which gives us \(2(9x^{2} - 16)\).

Step 4: Identify the difference of squares

The expression inside the parenthesis \((9x^{2} - 16)\) is a difference of squares, which could be further factored.

Step 5: Factor the difference of squares

The difference of squares can be factored into \((3x - 4)(3x + 4)\). Thus, the final factored expression is \(2(3x - 4)(3x + 4)\).

Key Concepts

Common Factor

Factoring by the common factor is a key technique in simplifying polynomial expressions. A common factor is a number or variable that divides all terms in a polynomial without leaving a remainder. To find it:

• Look at each term separately. For example, in the terms -32 and 18x², examine their numerical coefficients and variables.
• Calculate the greatest common factor (GCF). For the example, the GCF of -32 and 18x² is 2, since both -32 and 18 are divisible by 2.
• Factor out the common factor from the polynomial. This means you will divide each term by the GCF and express the polynomial as a product of the GCF and a simplified polynomial. For instance, -32 + 18x² becomes 2(-16 + 9x²) after factoring out 2.

Finding a common factor is often the first step in a longer factoring process, helping to simplify the expression for further manipulation.

Difference of Squares

The difference of squares is a specific type of polynomial expression that can be easily factored using a simple formula. A difference of squares looks like this:

• It has exactly two terms.
• The terms must be squared numbers or variables.
• It involves a subtraction between these terms, for example, a² - b².

To factor a difference of squares expression, apply the formula \[(a^2 - b^2) = (a - b)(a + b)\]. For instance, in the polynomial 9x² - 16, recognize 9x² as (3x)² and 16 as 4². Therefore, \[(9x^2 - 16) = (3x - 4)(3x + 4)\]. This method relies on understanding how to identify squares and perform basic arithmetic operations, making it a powerful tool in algebra.

Polynomial Expressions

Polynomial expressions consist of terms made up of variables raised to positive integer powers and coefficients. They can be as simple as a single number or variable, or as complex as multivariable equations with many terms. Understanding them involves:

• Identifying terms and their components - like coefficients and powers. For example, in the term 18x², 18 is the coefficient, and x² signifies that x is squared.
• Recognizing the degree of a polynomial - which is the highest power of the variable in the expression. In 18x², the degree is 2.
• Learning how to perform operations like addition, subtraction, multiplication, and factoring.

These expressions are fundamental in algebra and serve as the building blocks for more complex mathematical operations. Mastering the properties and operations of polynomial expressions is essential for solving equations and modeling real-world scenarios.