Problem 39 - Elementary Algebra

Question

Factor the greatest common factor from each polynomial. \(-2 x-4\)

Step-by-Step Solution

Answer

The factored polynomial is \(-2(x + 2)\).

Step 1: Identify the terms

The given polynomial is \(-2x - 4\). Identify the terms in the polynomial, which are \-2x\ and \-4\.

Step 2: Find the GCF

Determine the greatest common factor (GCF) of the coefficients from each term. The coefficients are \(-2\) and \(-4\). The GCF of \(-2\) and \(-4\) is \(-2\).

Step 3: Factor out the GCF

Factor out the GCF, \(-2\), from each term in the polynomial. This gives \(-2(x + 2)\).

Step 4: Simplify the expression

After factoring out the GCF, the simplified expression is \(-2(x + 2)\). Make sure to check the result by distributing \(-2\) back into the polynomial to verify correctness.

Key Concepts

Greatest Common Factor

When factoring polynomials, the greatest common factor (GCF) is crucial. The GCF is the largest factor that can divide each term of the polynomial without leaving a remainder.

For the polynomial \(-2x - 4\), we need to identify the GCF of \(-2\) and \(-4\). The GCF here is \(-2\) because it can divide both terms completely:

• \(-2 \div -2 = 1 \)
• \(-4 \div -2 = 2 \)

This process simplifies the polynomial by reducing it to factors common to all terms.

Polynomial Terms

A polynomial is made up of terms, which are the building blocks of the expression. Terms can include variables, coefficients, and exponents. In the polynomial \(-2x - 4\), there are two terms:

• \(-2x\) which is a variable term with coefficient \(-2\)
• \(-4\) which is a constant term

Identifying the terms helps in understanding the structure of the polynomial and in applying the correct steps to factor it.

Factoring Process

The factoring process is a method used to simplify polynomials. Here are the steps again with detailed explanations:

1. **Identify the Terms**: Recognize each term in the polynomial. Here, the terms are \(-2x\) and \(-4\).
2. **Find the GCF**: Determine the greatest common factor of all the terms. For this example, the GCF is \(-2\).
3. **Factor Out the GCF**: Rewrite each term as a product of the GCF and another factor. This means expressing \(-2x - 4\) as \(-2(x + 2)\).
4. **Simplify the Expression**: Ensure the polynomial is in its simplest form. To verify, distribute the GCF back: \(-2(x + 2) = -2x - 4 \).

Simplifying Expressions

Simplifying expressions makes them easier to work with and understand. When you factor out the GCF, you reduce a polynomial to its simplest form.

In the context of our example, after factoring out \(-2\) from \(-2x - 4\), the polynomial becomes \(-2(x + 2)\). This form is simpler and more manageable:

• It's shortened, making it easier to use in further calculations.
• It clearly shows the relationships and factors in the polynomial.

Simplification is a key skill in algebra to help make problem-solving faster and less error-prone.