Problem 22 - Elementary Algebra

Question

Factor completely. $$ 20 x 3-45 x $$

Step-by-Step Solution

Answer

The completely factored form is \( 5x(2x - 3)(2x + 3) \).

Step 1: Identify Common Factors

First, examine each term to identify any common factors. The expression is \( 20x^3 - 45x \). Both terms have a common factor of \( 5x \).

Step 2: Factor Out the Greatest Common Factor

Factor out the greatest common factor (GCF), which is \( 5x \), from the expression. This gives:\[ 20x^3 - 45x = 5x(4x^2 - 9) \]

Step 3: Recognize a Difference of Squares

The expression inside the parentheses, \( 4x^2 - 9 \), is a difference of squares. It can be written as \((2x)^2 - 3^2\).

Step 4: Apply the Difference of Squares Formula

Factor \( 4x^2 - 9 \) using the difference of squares formula, \( a^2 - b^2 = (a - b)(a + b) \). Here, \( a = 2x \) and \( b = 3 \), so:\[ 4x^2 - 9 = (2x - 3)(2x + 3) \]

Step 5: Combine and Write the Completely Factored Form

Combine the factors obtained to write the completely factored form of the original expression:\[ 20x^3 - 45x = 5x(2x - 3)(2x + 3) \]

Key Concepts

Greatest Common Factor

When we factor polynomials, the first thing to check for is the Greatest Common Factor (GCF). The GCF is the largest expression that can be factored out of each term in the polynomial.
This simplifies the expression and makes further factorization manageable. To find the GCF in the expression \(20x^3 - 45x\):

• Look for the highest number that divides evenly into the coefficients (numbers in front) of each term. For \(20\) and \(45\), the highest common divisor is \(5\).
• Next, identify the common variables. Here, both terms include \(x\), and the smallest power of \(x\) is \(x^1\).

Combining these, the GCF for this expression is \(5x\). By factoring \(5x\) out, the expression reduces to \(5x(4x^2 - 9)\).
This step simplifies the polynomial and prepares it for other techniques like the difference of squares.

Difference of Squares

The difference of squares is a common structure in algebra that enables specific polynomial expressions to be factored easily. It relates to expressions where you have a subtraction between two perfect squares.
These take the form \(a^2 - b^2\), and can always be factored as \((a - b)(a + b)\). In the factored expression \(4x^2 - 9\):

• Recognize \(4x^2\) as \((2x)^2\) and \(9\) as \(3^2\). So \(4x^2 - 9\) is a difference of squares, \((2x)^2 - (3)^2\).
• Apply the formula, letting \(a = 2x\) and \(b = 3\).

This leads to the factorization: \((2x - 3)(2x + 3)\).
Understanding and identifying a difference of squares can simplify complicated expressions swiftly, saving time and effort in solving algebraic equations.

Algebraic Expressions

Algebraic expressions contain numbers, variables, and operations. They are the fundamental building blocks in algebra.
Factoring is a technique used to simplify algebraic expressions into their products by uncovering structures like common factors or special patterns such as the difference of squares.For instance, given the expression \(20x^3 - 45x\):

• We started by factoring out the GCF, \(5x\), reducing the complexity of the expression.
• Next, we identified and applied the difference of squares to further factor \(4x^2 - 9\) into \((2x - 3)(2x + 3)\).

Ultimately, the expression is rewritten in a completely factored form as \(5x(2x - 3)(2x + 3)\).
Through effective manipulation of algebraic expressions like this, many mathematical problems become more straightforward and tractable.