Problem 41 - Elementary Algebra

Question

Factor. $$ 5 x 3-65 x 2+60 x $$

Step-by-Step Solution

Answer

The expression factored is \(5x(x - 12)(x - 1)\).

Step 1: Identify Common Factors

Examine the expression \(5x^3 - 65x^2 + 60x\) to see if there is any common factor for all the terms. Notice that each term is a multiple of \(5x\).

Step 2: Factor Out the Greatest Common Factor

Factor \(5x\) out of the expression: \[ 5x(x^2 - 13x + 12) \]

Step 3: Factor the Quadratic Expression

The quadratic \(x^2 - 13x + 12\) needs to be factored. Look for two numbers that multiply to \(12\) and add up to \(-13\). These numbers are \(-12\) and \(-1\).

Step 4: Express the Quadratic in Factored Form

Rewrite \(x^2 - 13x + 12\) as \((x - 12)(x - 1)\) since \(-12 + (-1) = -13\) and \((-12) \cdot (-1) = 12\).

Step 5: Combine the Factors

Combine all factors to express the original expression in its completely factored form: \[ 5x(x - 12)(x - 1) \]

Key Concepts

Understanding the Greatest Common Factor

Finding the greatest common factor (GCF) is an essential step in simplifying algebraic expressions. It's like finding the biggest number that can divide all terms in a polynomial without leaving a remainder. In the expression \(5x^3 - 65x^2 + 60x\), we notice that each term has a common factor of \(5x\).
This means:

• \(5x^3\) becomes \(5x \cdot x^2\)
• \(-65x^2\) becomes \(-5x \cdot 13x\)
• \(60x\) becomes \(5x \cdot 12\)

By factoring out \(5x\), we simplify the polynomial, preparing it for further factoring. Identifying the GCF makes the next steps more straightforward.

Delving into Quadratic Expressions

A quadratic expression is a polynomial of degree two. It typically appears in the form \(ax^2 + bx + c\). In our exercise, the quadratic expression is \(x^2 - 13x + 12\).
Key characteristics include:

• The quadratic term \(x^2\), responsible for the parabolic shape when graphed.
• The linear term \(-13x\), influencing the direction and steepness.
• The constant term \(12\), which shifts the graph vertically.

Quadratic expressions are central to algebra and often require factoring to solve for roots or zeroes.

Exploring the Factoring Process

The factoring process involves breaking down a complex expression into simpler components. These components or 'factors' can multiply to give the original expression. For \(x^2 - 13x + 12\), we look for two numbers that:

• Multiply to \(12\)
• Add up to \(-13\)

The numbers \(-12\) and \(-1\) fit these criteria perfectly.
Thus, the expression becomes \((x - 12)(x - 1)\), shaping it into a product of two binomials. This method of factoring quadratics is sometimes called "factoring by inspection" and is a useful technique when the quadratic is easily factorable.

Grasping Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and mathematical operations like addition or multiplication. They form the foundation of algebra, allowing us to model and solve real-world problems. The exercise starts with \(5x^3 - 65x^2 + 60x\), a polynomial involving:

• A cubic term \(5x^3\)
• A quadratic term \(-65x^2\)
• A linear term \(60x\)

These terms align based on decreasing powers of \(x\).
Factoring simplifies these expressions, making them easier to manipulate and solve.