Problem 39 - Elementary and Intermediate Algebra

Question

Factor expression. Factor out any GCF first. \(3 x^{3}-243 x\)

Step-by-Step Solution

Answer

The factored form is \(3x(x + 9)(x - 9)\).

Step 1: Identify the Greatest Common Factor (GCF)

Examine the expression \(3x^3 - 243x\) to identify the greatest common factor of all terms. Both terms have a common factor of 3 and an \(x\). Hence, the GCF is \(3x\).

Step 2: Factor Out the GCF

Divide each term in the expression \(3x^3 - 243x\) by the GCF \(3x\): - \(3x^3 \div 3x = x^2\)- \(243x \div 3x = 81\)Thus, factoring out the GCF gives: \(3x(x^2 - 81)\).

Step 3: Recognize a Difference of Squares

The expression inside the parentheses, \(x^2 - 81\), is a difference of squares, since \(x^2 - 81 = x^2 - 9^2\). A difference of squares can be factored using the formula \(a^2 - b^2 = (a+b)(a-b)\).

Step 4: Factor the Difference of Squares

Apply the difference of squares formula to \(x^2 - 9^2\):\(x^2 - 9^2 = (x + 9)(x - 9)\).Therefore, the completely factored form of the original expression is \(3x(x + 9)(x - 9)\).

Key Concepts

Greatest Common Factor

The Greatest Common Factor (GCF) is a key concept in simplifying expressions in algebra. It refers to the largest factor that is common to all terms in a polynomial. To find the GCF in an expression like \(3x^3 - 243x\), we first need to identify the common numerical and variable components of each term.
In this case, the following needs to be checked:

• Numerical Factors: Look at the coefficients, which are 3 and 243. The number 3 is the greatest number that divides both 3 and 243 evenly.
• Variable Factors: Examine the variable portion. Both terms include the variable \(x\), with the lowest power being \(x^1\). Hence, \(x\) is also a common factor.

Combine these to find that the GCF is \(3x\). By factoring this out of the polynomial, we obtain a simpler expression: \(3x(x^2 - 81)\). This step helps in breaking down the polynomial, making further simplification manageable.

Difference of Squares

The difference of squares is a specific algebraic formula used to factor certain types of expressions. It applies when you have a difference between two perfect squares. In simple terms, perfect squares are numbers like \(x^2\) and \(9^2\) because they result from squaring a term.
In the expression \(x^2 - 81\), we can see there are two perfect squares:

• \(x^2\), which is \((x)^2\)
• 81, which can be rewritten as \(9^2\)

The pattern \(a^2 - b^2 = (a+b)(a-b)\) is crucial for factoring the difference of squares. We use this pattern by assigning \(a = x\) and \(b = 9\). Hence, \(x^2 - 9^2 = (x+9)(x-9)\).
Applying this formula is a powerful trick to simplify expressions and often appears in more advanced algebra problems. The original expression \(3x(x^2 - 81)\) thus becomes \(3x(x + 9)(x - 9)\), its fully factored form.

Algebraic Expressions

Understanding algebraic expressions is fundamental in solving and simplifying equations. They are combinations of variables, numbers, and operations like addition, subtraction, multiplication, and division.
For instance, the expression \(3x^3 - 243x\) consists of:

• Terms: Parts of the expression separated by addition or subtraction, in this case, \(3x^3\) and \(-243x\).
• Coefficients: Numerical factors that multiply a variable, here 3 and 243.
• Variables: Symbols, like \(x\), used to represent unknown values or quantities.
• Exponents: Power to which a number or variable is raised, for example, the 3 in \(x^3\).

An important part of working with algebraic expressions is rewriting them in simpler forms through factoring. By removing common factors and applying special rules like the difference of squares, expressions can be transformed into their elemental components. This not only clarifies the mathematical relationships within the expression but also paves the way for solving more complex algebraic equations in future problems.