Problem 25 - Algebra 1: Concepts and Skills

Question

Find the greatest common factor of the terms and factor it out of the expression. \(18 d^{6}-6 d^{2}+3 d\)

Step-by-Step Solution

Answer

The greatest common factor of the terms in the expression is \(3d\), and the expression factored out is \(3d(6d^{5} - 2d + 1)\).

Step 1: Determine the GCF of the terms

In this expression, the terms are \(18d^{6}\), \(-6d^{2}\), and \(3d\). We identify the common factors of the three terms. Here, the common factors are 3 and \(d\). Therefore, the greatest common factor (GCF) is \(3d\). This means that all the terms in the expression can be divided evenly by \(3d\).

Step 2: Factor out the GCF from each term

Next, divide each term in the original expression by the GCF to determine the remaining factors in each term. This leads to: \(18d^{6} ÷ 3d = 6d^{5}\), \(-6d^{2} ÷ 3d = -2d\), and \(3d ÷ 3d = 1\).

Step 3: Write the factored expression

The final step is to write the factored expression. The factored form includes the GCF, and the resulting expression when the original expression is divided by the GCF. Therefore, the GCF of \(3d\) factored out of the original expression gives us \(3d(6d^{5} - 2d + 1)\).

Key Concepts

Greatest Common Factor

Understanding the greatest common factor (GCF) is essential in simplifying algebraic expressions. The GCF of a set of terms is the largest factor that all terms have in common. Think of it like finding the biggest piece of a puzzle that fits into all other pieces. In the exercise \(18d^{6}-6d^{2}+3d\), to identify the GCF, we first list the factors for each term. For \(18d^{6}\), \(6d^{2}\), and \(3d\), the number 3 and the variable \(d\) appear as factors in each term. Hence, \(3d\) is the GCF.

Why does the GCF matter? It helps in breaking down expressions into simpler forms, which can be particularly useful for further algebraic operations, such as polynomial division or solving equations. It’s like tidying up an equation by removing common clutter, leaving a more organized and often easier problem to work with.

Algebraic Expressions

Algebraic expressions are like sentences in the language of mathematics, composed of numbers, variables, and operations such as addition and subtraction. They are the building blocks of algebra and are used to represent real-world situations. When we factor an algebraic expression, we are essentially decompressing the sentence, breaking it into more manageable parts, or finding a way to express it more succinctly. In the example, \(18d^{6}-6d^{2}+3d\) is simplified by factoring out the GCF, resulting in \(3d(6d^{5} - 2d + 1)\). This transformation is invaluable as it makes it easier to understand and manipulate the expression for solving equations or inequalities.

Why Factor Algebraic Expressions?

• To simplify calculations and reduce complexity
• To solve equations more easily
• To find roots or solutions to equations
• To understand the properties of the expression better

Learning to factor algebraic expressions is a stepping stone to mastering algebra and is vital for progressing to more advanced mathematical concepts.

Polynomial Division

Polynomial division, similar to long division with numbers, is a technique used to divide a polynomial by another polynomial. For example, when factoring out the GCF in an expression, we are actually performing a simple form of polynomial division. Dividing \(18d^{6}\), \(6d^{2}\), and \(3d\) by the GCF \(3d\) is exactly that. The operation helps us in reducing the polynomial to a simpler form, \(6d^{5} - 2d + 1\), and is particularly useful when dealing with more complex polynomial expressions.

Excelling at polynomial division is key to anyone's toolkit in algebra. It enables students to tackle problems ranging from function simplification to finding solutions to polynomial equations. Grasping this concept opens doors to understanding even more advanced areas like calculus, where polynomial division becomes an everyday tool.