Problem 45 - Elementary Algebra

Question

Factor out the GCF. $$ 9 x 4+18 x 3-3 x 2 $$

Step-by-Step Solution

Answer

The expression factored is \(3x^2(3x^2 + 6x - 1)\).

Step 1: Identify the GCF

Examine each term in the expression: \(9x^4\), \(18x^3\), and \(3x^2\). Identify the greatest common factor among the coefficients (9, 18, 3) and the variables (\(x^4, x^3, x^2\)). The GCF of the coefficients is 3, and for the variable part, it is the highest power of \(x\) common to all terms, which is \(x^2\). Therefore, the GCF is \(3x^2\).

Step 2: Divide Each Term by the GCF

Divide each term of the expression by the GCF \(3x^2\). Calculating these: \(\frac{9x^4}{3x^2} = 3x^2\), \(\frac{18x^3}{3x^2} = 6x\), and \(\frac{-3x^2}{3x^2} = -1\).

Step 3: Write the Factored Expression

The original expression can be rewritten as \(GCF \times \, \text{resulting expression from Step 2}.\) Thus, it becomes \(3x^2(3x^2 + 6x - 1)\).

Key Concepts

Greatest Common Factor (GCF)

In algebra, the Greatest Common Factor (GCF) is the largest factor shared by two or more numbers or terms. Factoring polynomials often starts with identifying the GCF to simplify expressions. The process involves looking at both the numerical coefficients and any variables involved.
Here's how we find it:

• **Coefficients**: Determine the highest number that evenly divides each of the coefficients. In the exercise above, the coefficients are 9, 18, and -3. The greatest number that divides all of them is 3.
• **Variables**: Identify the lowest power of the variable common to each term. For the variables in the expression, notice they all contain at least an \(x^2\).

Combining both results gives us a GCF of \(3x^2\). This factor is pivotal in simplifying polynomial expressions by dividing it out from each term.

Polynomial Expressions

Polynomial expressions comprise variables (denoted usually by symbols like \(x\)) and coefficients, which are constants. These expressions can have one or more terms connected via addition or subtraction. Each term is essentially a monomial.In practice, polynomial expressions can be:

• **Single-Term (Monomial)**: E.g., \(3x^2\)
• **Two-Terms (Binomial)**: E.g., \(x^3 - 4x\)
• **Three-Terms (Trinomial)**: E.g., \(x^2 + 3x + 2\)

Polynomials of higher complexity, called multinomials, may consist of four or more terms.
In our exercise, the expression \(9x^4 + 18x^3 - 3x^2\) is a trinomial. Recognizing the type of polynomial is critical for choosing the best strategy for factoring.

Algebraic Expressions

Algebraic expressions make up the foundation of algebra and consist of numbers, symbols, and operators indicative of a mathematical scenario. These expressions can range from simple arithmetic operations to complex polynomials, involving:

• **Variables**: Symbols like \(x\) represent unknown quantities.
• **Constants**: Known numbers, such as 3 or -1, within the expression.
• **Operators**: Mathematic symbols like +, -, ×, and ÷.

Understanding algebraic expressions means knowing how to manipulate them using operations like addition, subtraction, and factoring.
Factoring is particularly key when trying to simplify or solve equations. In our example, factoring out the GCF turned \(9x^4 + 18x^3 - 3x^2\) into \(3x^2(3x^2 + 6x - 1)\), thus simplifying the expression for further operations or solutions.