Problem 44 - Algebra 1: Concepts and Skills

Question

Factor the expression completely. \(2 x^{3}-6750\)

Step-by-Step Solution

Answer

The expression \(2 x^{3}-6750\) can be factored completely as \(2(x^3-3375)\).

Step 1: Identify the GCF

Looking at the constants, the GCF of 2 and 6750 is 2. Considering the variable part which is \(x^3\) in the first term, it's evident that there's no variable in the second term. As such, the GCF will be the common factor in their numerical coefficients, which is 2.

Step 2: Divide the entire polynomial by the GCF

Now divide each term in the original equation by 2, which is the GCF identified earlier: \(2 x^{3}/2 = x^{3}\), and \(-6750/2 = -3375\).

Step 3: Write down the factored form

The factored polynomial is thus given as the product of the GCF and the result of dividing each term in the original polynomial by the GCF. Hence, the factored form of the polynomial becomes: \(2(x^3-3375)\).

Key Concepts

Greatest Common Factor

Finding the greatest common factor (GCF) of numbers is like finding the biggest number that can evenly divide each of them. In our exercise, the polynomial is made up of two numbers: 2 and 6750. When we talk about the GCF here, we are looking for the largest number that divides both. In this case, it is 2 because:

• 2 can divide 2 itself.
• 2 can also divide 6750.

This process helps us simplify polynomials by pulling out common factors first. The GCF is crucial in factoring because it allows us to break down larger expressions into simpler parts. By finding the GCF, we streamline the next steps in factoring, making it easier to manage often complex expressions.

Polynomial Division

Once the GCF is identified, we use polynomial division to simplify the expression. The operation involves dividing each term in the polynomial by the GCF. For our polynomial, this means:

• The first term is \(2x^3\), and when we divide it by 2, we get \(x^3\).
• The second term is \(-6750\), and dividing that by 2 gives us \(-3375\).

This step reduces the original polynomial into a simpler one, \(x^3 - 3375\). It's important because it prepares the polynomial for further factoring or solving, if needed. By systematically dividing each term, we ensure every part of the polynomial is accounted for, maintaining a balanced equation.

Factoring by GCF

The final step in our process is called "factoring by GCF." This involves expressing the original polynomial as a product of the GCF and the polynomial left after division. After we've divided the polynomial terms, we rewrite the initial expression as:

• \(2(x^3 - 3375)\)

This means the polynomial \(2x^3 - 6750\) can be viewed as 2 times \(x^3 - 3375\). This step is vital because it fully simplifies the expression while preserving its value. Factoring by GCF also makes it easier to handle the polynomial in equations or further operations, like solving for roots or graphing. This technique is a fundamental aspect of simplifying expressions in algebra.