Problem 1 - Algebra 2

Question

Factor completely. If the polynomial is not factorable, write prime. $$ -12 x^{2}-6 x $$

Step-by-Step Solution

Answer

The polynomial factors to \(-6x(x + 1)\).

Step 1: Identify the Greatest Common Factor (GCF)

Look for the greatest common factor in the terms of the polynomial \(-12x^2 - 6x\). The GCF here is \(-6x\).

Step 2: Factor Out the GCF

Factor out the GCF from each term in the polynomial. The expression becomes:\(-6x(x + 1)\).

Key Concepts

Greatest Common Factor

Finding the greatest common factor (GCF) is the first step when trying to factor polynomials. The GCF is the largest factor that divides each term in the expression. For instance, in the polynomial -12x^2 - 6x, we need to break down each term:

• -12x^2 can be broken into -6 * 2 * x * x.
• -6x is already -6 * x.

Both terms have -6 and x in common. Therefore, the GCF of the polynomial is -6x.
Understanding how to determine the GCF is essential, as it simplifies the polynomial and makes further factoring feasible. It's important to identify and apply this concept as the initial step in many algebraic expressions.

Polynomial Expressions

Polynomial expressions are sums of terms, where each term consists of a variable raised to a non-negative integer power multiplied by a coefficient. The expression -12x^2 - 6x is an example of a polynomial. It has two terms: -12x^2 and -6x.
Key points about polynomial expressions include:

• The degree of a polynomial is determined by the highest exponent of the variable.
• Terms are separated by addition or subtraction.
• Each term consists of a coefficient and a variable base.

Recognizing and working with polynomials is a fundamental skill in algebra. They appear in various forms and can become more complex, but they often can be broken down to simpler forms when factoring.

Factoring Techniques

Factoring techniques are strategies used to express a polynomial as a product of its factors. These techniques are very useful for simplifying expressions and solving equations.
The process of factoring usually starts with finding the GCF, like in: -12x^2 - 6x, where we identify the GCF as -6x. After identifying the GCF, the expression is rewritten by dividing each term by this factor.
-12x^2 / -6x = 2x and -6x / -6x = 1. This simplifies the expression to -6x(x + 1).
Some common factoring techniques include:

• Factoring by grouping.
• Factoring trinomials.
• Using special formulas like the difference of squares.

Mastery of these techniques builds a foundation for more advanced topics in math, such as quadratic equations and polynomial division. The ability to simplify and manipulate polynomials through factoring is continuously useful in algebraic problem-solving.