Problem 5 - College Algebra Essentials

Question

$$\text { Factor out the greatest common factor.}$$ $$9 x^{4}-18 x^{3}+27 x^{2}$$

Step-by-Step Solution

Answer

The factored form of the expression \(9x^{4} - 18x^{3} + 27x^{2}\) is \(9x^{2}(x^{2} - 2x + 3)\)

Step 1: Identify the Greatest Common Factor

To find the Greatest Common Factor (GCF) of the numerical coefficients and variables in each term, we need to find the highest power of \(x\) common to all terms and the largest number that divides evenly into all coefficients.\n\nLooking at the numbers, the coefficients are 9, 18, and 27. The number 9 divides evenly into all these coefficients.\n\nFor the variables, we notice that every term has an \(x\) raised to a certain power. The smallest power is 2. Therefore, the GCF of \(x\) in all the terms is \(x^{2}\).\n\nCombining these, the GCF of the entire expression is \(9x^{2}\).

Step 2: Factor Out the GCF

Once the GCF has been identified, the next step is to divide each term in the polynomial by the GCF. This leaves us with the remaining factors:\n\n\[\frac{{9x^{4}}}{{9x^{2}}} - \frac{{18x^{3}}}{{9x^{2}}} + \frac{{27x^{2}}}{{9x^{2}}}\]\n\nThis simplifies to: \(x^{2} - 2x + 3\).

Step 3: Write Final Answer

The factored form of the original expression is the GCF multiplied by the result after factoring out the GCF:\n\n\(9x^{2}(x^{2} - 2x + 3)\)

Key Concepts

Greatest Common Factor

Understanding the 'greatest common factor' (GCF) is crucial when simplifying algebraic expressions. The GCF of a set of numbers or algebraic terms is the largest factor that divides each of them without leaving a remainder.

When factoring polynomials, identifying the GCF of all terms allows you to simplify the expression by pulling out commonalities. This is similar to sharing a pie evenly among friends; you want to find the largest slice size (factor) that you can give to each person (term) without having any pie left (remainder).

In the polynomial 9x^{4}-18x^{3}+27x^{2}, for instance, the GCF is identified by looking at both the coefficients and the variables separately. The number 9 is the highest numerical coefficient that can divide 9, 18, and 27 evenly. Simultaneously, x^{2} is the highest power of x present in every term. Therefore, the overall GCF is 9x^{2}.

Algebraic Expressions

Algebraic expressions are combinations of constants, variables, and operations like addition, subtraction, multiplication, and division. In our exercise, 9x^{4}-18x^{3}+27x^{2} is an example of an algebraic expression made up of three terms.

Each term is a product of a coefficient (a constant like 9, 18 or 27) and a variable raised to a power (like x^{2}, x^{3}, or x^{4}). These expressions can represent complex relationships and are the basis for solving algebra problems. Understanding how to manipulate them through operations such as factoring is a fundamental skill in algebra.

Polynomial Division

Polynomial division is used when factoring an expression to simplify each term by the GCF. It involves dividing each term of the polynomial by the same divisor - the GCF. In our case, this step breaks down the larger polynomial into a more manageable expression.

Each term of the polynomial 9x^{4}-18x^{3}+27x^{2} is divided by the GCF 9x^{2}. The division results in a simpler expression, x^{2} - 2x + 3, where each term is a factor of the original terms. This step is similar to reducing fractions to their simplest form. By dividing the polynomial by the GCF, you're essentially 'simplifying the expression' to its core components.

Simplifying Expressions

The process of simplifying expressions requires combining like terms and factoring out common factors to reduce the complexity of the expression. Once the GCF is factored out, the remaining expression should have no common factors other than 1. This step is essential since it often leads to a form that is easier to use for further operations in algebra, such as solving equations.

In our exercise, the initial polynomial was simplified by factoring out the GCF, resulting in the final factored expression 9x^{2}(x^{2} - 2x + 3). Each step taken to reach this point involves crucial algebraic principles that serve to streamline the expression and enhance our understanding of its structure.