Problem 85 - College Algebra with Modeling and Visualization

Question

Factor the expression. \(9 y^{3}+30 y^{2}+25 y\)

Step-by-Step Solution

Answer

The factored form is \(y(3y + 5)^2\).

Step 1: Identify the Greatest Common Factor (GCF)

Start by identifying the greatest common factor (GCF) of all terms in the expression. The terms are \(9y^3\), \(30y^2\), and \(25y\). The common factor in all terms is \(y\), so the GCF is \(y\).

Step 2: Factor the GCF from the Expression

Factor out the GCF \(y\) from each term of the expression. The expression becomes: \[y(9y^2 + 30y + 25)\].

Step 3: Recognize the Quadratic Form

Look at the quadratic expression \(9y^2 + 30y + 25\) inside the parentheses. This expression is in the standard form \(ax^2 + bx + c\), where \(a = 9\), \(b = 30\), and \(c = 25\).

Step 4: Use the Quadratic Trinomial Factoring Method

Since \(9y^2 + 30y + 25\) is a perfect square trinomial, notice that this trinomial equals \((3y + 5)^2\). This is because \[(3y + 5)(3y + 5) = 9y^2 + 15y + 15y + 25 = 9y^2 + 30y + 25.\]

Step 5: Write the Fully Factored Expression

Combine the results from Steps 2 and 4. The expression \(9y^3 + 30y^2 + 25y\) fully factors to \[y(3y + 5)^2\].

Key Concepts

Quadratic Expressions

Quadratic expressions form a fundamental building block in algebra. These expressions are typically of the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(x\) represents the variable. The highest degree of any term in a quadratic expression is two, characterized by the \(x^2\) term.

Understanding the structure of a quadratic expression is crucial for factoring and solving equations. When you see a quadratic expression, take note of the coefficients \(a\), \(b\), and \(c\). Each plays a specific role in both the shape of the parabola represented by the quadratic equation and the methods used for factoring and solving it.

• \(a\) determines the direction and width of the parabola
• \(b\) affects the symmetry and position of the vertex
• \(c\) represents the y-intercept and impacts the entire graph's vertical position.

Identifying these components is the first step towards tackling more complex quadratic problems, including perfect square trinomials.

Greatest Common Factor

The greatest common factor (GCF) is a valuable tool in simplifying expressions and factoring polynomials. It refers to the highest factor that divides each term of the polynomial without leaving a remainder. The GCF is critical when you're first breaking down polynomial expressions into simpler parts, often used in the initial step of any factoring process.

To determine the GCF, list the factors of each term in the polynomial. For instance, the polynomial \(9y^3 + 30y^2 + 25y\) has the terms \(9y^3\), \(30y^2\), and \(25y\). The common factor among these terms is \(y\), making \(y\) the GCF.

• Factor each term: \(9y^3 = 9 \cdot y \cdot y^2\), \(30y^2 = 30 \cdot y\), and \(25y = 25 \cdot y\).
• Identify the smallest powers of all common variables across the terms: In our example, it is \(y^1\).

Using the GCF streamlines the expression, reducing it to simpler, more manageable terms, like slicing down overgrown branches to reveal the core structure of a tree.

Perfect Square Trinomial

A perfect square trinomial is a special type of quadratic expression. It can be expressed as the square of a binomial, making it especially easy to factor. Identifying perfect square trinomials allows you to quickly write them as \((a + b)^2\), a process that can significantly speed up your problem-solving.

For example, in the polynomial \(9y^3 + 30y^2 + 25y\), once you factor out the GCF \(y\), you are left with the quadratic \(9y^2 + 30y + 25\). This expression is recognized as a perfect square trinomial because it can be rewritten as \((3y + 5)^2\).

• Check if the first and last terms are perfect squares: Here, \(9y^2\) is \((3y)^2\) and \(25\) is \(5^2\).
• Ensure that twice the product of the square roots of these terms equals the middle term: \(2 \times 3y \times 5 = 30y\).

These simple checks help confirm that you are dealing with a perfect square trinomial, making it straightforward to rewrite and factor efficiently.