Problem 58 - Beginning and Intermediate Algebra

Question

Factor completely, if possible. Begin by asking yourself, "Can I factor out a GCF?" $$6 g^{5}+6 g^{4}-12 g^{3}$$

Step-by-Step Solution

Answer

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

Step 1: Identify the GCF

For the given expression \(6g^5+6g^4-12g^3\), the greatest common factor (GCF) is the highest power of the common factor, which we can see is \(6g^3\).

Step 2: Factor out the GCF

We will now factor out the GCF from the given expression: \(6g^5+6g^4-12g^3 = 6g^3(g^2 + g - 2)\)

Step 3: Check for further factoring

Now, we have factored out the GCF, but we need to check if the remaining expression \((g^2 + g - 2)\) can be factored further. Using trial and error, we can find that it can be factored as follows: \((g^2 + g - 2) = (g + 2)(g - 1)\) So, our fully factored expression is: \(6g^5+6g^4-12g^3 = 6g^3(g^2 + g - 2) = 6g^3(g + 2)(g - 1)\)

Key Concepts

Greatest Common Factor

The greatest common factor, or GCF, is a key concept in factoring polynomials. It refers to the largest factor shared by all terms in a polynomial expression. To find the GCF, you

• Identify the smallest power of each variable present in the terms.
• Look for the smallest coefficient that can divide each term.

In our example, we deal with the polynomial expression \(6g^5 + 6g^4 - 12g^3\). Each term has a coefficient (number in front of the variable) and a variable part (\(g\) raised to some power). Here, all the coefficients (6, 6, and 12) can be evenly divided by 6, which makes 6 the GCF of the numbers.
Next, observe the variables. The smallest power of \(g\) common to all terms is \(g^3\).
This gives us the GCF for the polynomial as \(6g^3\). By factoring out \(6g^3\), we simplify the polynomial, making further factoring easier. Begin any factoring by asking, "Can I find a GCF?"

Factoring Quadratic Expressions

After pulling out the GCF, a quadratic expression might remain, as it did in our example: \(g^2 + g - 2\). Factoring quadratic expressions involves rewriting the quadratic in a product of two binomials when possible. You often use a method called 'trial and error' or 'factoring by inspection' to find binomials that satisfy two conditions:

• Their product must equal the original quadratic expression
• They must expand back to the same expression when multiplied

For \(g^2 + g - 2\), our goal is to find two numbers that multiply to \(-2\) (constant term) and add to \(1\) (coefficient of \(g\)). Testing different pairs, we find \(g + 2\) and \(g - 1\) satisfy these conditions. A quick expansion verifies this: \[(g + 2)(g - 1) = g^2 - g + 2g - 2 = g^2 + g - 2\]
Using these techniques makes factoring quadratics systematic and gives confidence in finding solutions.

Polynomial Expressions

Working with polynomial expressions means dealing with terms that consist of variables raised to whole-number exponents and coefficients. In our context, polynomial expressions can often be simplified or decomposed into simpler factors.
Factoring allows you to express the polynomial as a product of its factors. This helps to reveal the roots, or solutions of the equation set to zero. The primary steps are:

• Finding the GCF and factoring it out
• Simplifying the expression by identifying patterns or using different factoring techniques, such as looking for a difference of squares or using the quadratic formula when applicable

The example of \(6g^5 + 6g^4 - 12g^3\) illustrates how to start with a complex expression and simplify it through logical steps. The key is organization and systematic problem-solving to handle even the toughest of polynomial challenges efficiently.