Problem 134 - College Algebra with Modeling and Visualization

Question

Factor the expression completely. \(18 x^{3}-60 x^{2}+50 x\)

Step-by-Step Solution

Answer

The expression factors to \(2x(3x - 5)^2\).

Step 1: Identify the Greatest Common Factor (GCF)

The expression given is \(18x^3 - 60x^2 + 50x\). Look at each term: \(18x^3\), \(-60x^2\), and \(50x\) to identify the common factors. The GCF for the numbers 18, 60, and 50 is 2. Each term also contains \(x\), so the overall GCF is \(2x\).

Step 2: Factor Out the GCF

Divide each term in the expression by the GCF \(2x\). The expression becomes: \[2x(9x^2 - 30x + 25)\].

Step 3: Factor the Quadratic Expression

Consider the quadratic expression \(9x^2 - 30x + 25\) inside the parentheses. We look for two numbers that multiply to \(9 \times 25 = 225\) and add to \(-30\). The two numbers are \(-15\) and \(-15\).

Step 4: Write the Quadratic as a Product of Binomials

Since the numbers found are identical \(-15\) and \(-15\), the quadratic \(9x^2 - 30x + 25\) can be rewritten into \((3x - 5)(3x - 5)\) or \((3x - 5)^2\).

Step 5: Write the Fully Factored Expression

Combine the above steps to express the fully factored form. The original expression \(18x^3 - 60x^2 + 50x\) is fully factored as \(2x(3x - 5)^2\).

Key Concepts

Factoring

Factoring is a technique in algebra where we break down an expression into a product of simpler expressions. Imagine you’re baking and want to divide a large dough into smaller, manageable pieces. Factoring works similarly, allowing us to simplify expressions for ease of manipulation and solving equations. This process involves recognizing patterns and applying known algebraic identities. Common techniques include taking out a common factor, grouping terms, or employing special formulas like the difference of squares. Practicing these strategies enhances your problem-solving skills and deepens your understanding of algebraic structures.

Greatest Common Factor (GCF)

The Greatest Common Factor (GCF) is the largest factor that divides two or more numbers. Think of it as finding the largest 'building block' that the numbers share. In the algebraic expression \(18x^3 - 60x^2 + 50x\), we identify and extract the GCF to simplify the expression. Here’s how:

• List the factors: Begin by finding the factors of each coefficient (18, 60, and 50). The largest common number is 2.
• Include variables: Since each term contains the variable \(x\), it is also a part of the GCF, resulting in \(2x\).

Factoring out the GCF is like peeling layers off a new concept, leaving us with a cleaner, more refined expression to work with.

Quadratic Expressions

Quadratic expressions are polynomial expressions of degree two, typically written in the form \(ax^2 + bx + c\). Dealing with quadratics involves various methods to simplify or solve such equations, especially factoring. For the expression we derived, \(9x^2 - 30x + 25\), we need numbers that multiply to the product of \(a\) and \(c\) (i.e., 225) and add up to \(b\) (i.e., -30). Once identified, these numbers help rewrite the middle term, allowing factorization by grouping. Some quadratics might even fit special patterns, like perfect square trinomials, which simplify further into squared binomials, as with \((3x - 5)^2\). Understanding these steps enables us to tackle quadratic equations with confidence, leading to efficient solutions in various algebraic problems.