Problem 9 - College Algebra

Question

In Exercises \(1-10\), factor out the greatest common factor. $$x^{2}(x-3)+12(x-3)$$

Step-by-Step Solution

Answer

The factored form of the given expression is \((x-3)(x^{2} + 12)\).

Step 1: Identify the GCF

The greatest common factor (GCF) is the largest expression that both terms have in common. Looking at the given expression \(x^{2}(x-3)+12(x-3)\), it can be seen that both terms contain \((x-3)\) so this is the GCF.

Step 2: Factor out the GCF

Next, factor out the GCF from each term in the original expression. This will result in the expression \((x-3)[x^{2} + 12]\).

Step 3: Simplify the Expression

Finally, simplify the expression as much as possible. In this case, the expression is already as simplified as it can be, so the solution is \((x-3)(x^{2} + 12)\).

Key Concepts

Greatest Common Factor

When solving polynomial expressions, one of the first techniques to apply is identifying the Greatest Common Factor (GCF). This concept refers to the largest expression that appears in each term of a polynomial. In simpler terms, it’s the biggest "piece" shared by all parts of your expression. Here's how to find it:

• Examine each term of the polynomial individually.
• Identify the common components, whether they are numbers, variables, or other expressions.
• Ensure it is the largest expression common to each term.

Let's take our example expression, \[x^{2}(x-3) + 12(x-3).\]Here, each part of the expression shows \((x-3)\)as a factor, making it the GCF. This method can simplify seemingly complicated polynomials right from the start, setting a clearer pathway for further operations.

Polynomial Expressions

A polynomial expression is a sum or difference of terms, each consisting of variables raised to a power, coefficients, or both. Understanding the structure of polynomial terms is key to handling them effectively. For instance:

• The expression \(x^{2}(x-3)\) contains terms formed from variables and numbers – here, involving powers of \(x\).
• The expression \(12(x-3)\) combines a constant with a variable part.

Polynomials can be simple, linear, or involve higher-degree terms. Recognizing these parts will help in deploying the right factoring techniques. In our example, recognizing that each part shares a common expression makes it easier to factor these polynomials in a structured manner.

Factoring Techniques

Factoring techniques are methods used to break down polynomials into products of simpler factors. This can simplify expressions and solve equations more easily. Here's a simplified approach:

• **Factoring out the GCF:** Start by factoring out the greatest common factor from the terms, which often simplifies the polynomial significantly.
• **Identifying patterns:** Look out for recognizable forms in the remaining polynomials like quadratics or differences of squares.

In our particular example, after factoring out the GCF \((x-3)\), we're left with \[(x-3)(x^{2} + 12).\]Here, \((x-3)\) is already factored out, simplifying the original expression significantly. Understanding and applying these techniques can turn complex expressions into manageable problems. It’s a crucial skill to master for algebraic manipulations.