Problem 86 - College Algebra with Modeling and Visualization

Question

Factor the expression. \(25 y^{3}-20 y^{2}+4 y\)

Step-by-Step Solution

Answer

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

Step 1: Identify the Greatest Common Factor (GCF)

First, look for the greatest common factor (GCF) among all the terms in the expression. The terms are: \(25y^3\), \(-20y^2\), and \(4y\). The GCF here is \(y\) because each term contains at least one \(y\).

Step 2: Factor out the GCF

Factor \(y\) out of the entire expression. Therefore: \(25y^3 - 20y^2 + 4y = y(25y^2 - 20y + 4)\).

Step 3: Rewrite the Quadratic Expression

Now, focus on factoring the quadratic expression inside the parenthesis: \(25y^2 - 20y + 4\). Remember, this is in the form of \(ax^2 + bx + c\), where \(a = 25\), \(b = -20\), and \(c = 4\).

Step 4: Factor the Trinomial by Grouping

To factor \(25y^2 - 20y + 4\), we need factors of \(25 \times 4 = 100\) that add to \(-20\). These are \(-10\) and \(-10\). Rewrite the middle term: \(25y^2 - 10y - 10y + 4\).

Step 5: Group Terms and Factor

Group terms in pairs: \((25y^2 - 10y) + (-10y + 4)\). Factor each pair: \(5y(5y - 2) - 2(5y - 2)\). Notice \(5y - 2\) is common in both groups.

Step 6: Factor Out the Common Binomial

Factor out the common binomial \((5y - 2)\): \((5y - 2)(5y - 2)\). Thus, \(25y^2 - 20y + 4 = (5y - 2)^2\).

Step 7: Combine with the GCF

Bring back the factored GCF from Step 2: \(y(5y - 2)^2\). So, the expression \(25y^3 - 20y^2 + 4y\) is fully factored as \(y(5y - 2)^2\).

Key Concepts

Greatest Common Factor

When you approach factoring polynomials, finding the greatest common factor (GCF) is often the first step. The GCF is the highest factor that evenly divides all terms in a given expression. Take the expression from the original exercise:

• The terms are: \(25y^3\), \(-20y^2\), and \(4y\).
• The highest common factor among these terms is \(y\), as each term contains at least one \(y\).

By factoring out this \(y\), you simplify the expression and make it easier to work with. This step reduces the given polynomial to \(y(25y^2 - 20y + 4)\), setting the stage for further factoring.

Trinomial Factoring

After isolating any common factors, your next target is often factoring a trinomial. Trinomials are polynomials with three terms, generally given in the form \(ax^2 + bx + c\). For our example, the expression inside the brackets, \(25y^2 - 20y + 4\), is a trinomial.

• The goal is to break this expression down into simpler binomial factors.
• We look for two numbers that multiply to \(a \times c\) (in this case, \(25 \times 4 = 100\)), and add to \(b\) (\(-20\)). Here, the numbers are \(-10\) and \(-10\).

Breaking down and rearranging the expression using these numbers helps to factor it through grouping.

Binomial Factoring

Factoring binomials involves expressing a polynomial as a product of two binomial expressions. After writing the middle term of the trinomial in terms of two numbers, you can proceed with binomial factoring.In our exercise, we wrote:\[ 25y^2 - 10y - 10y + 4 \]Next, group and factor these terms:

• Group: \((25y^2 - 10y) + (-10y + 4)\).
• Factor each pair: \(5y(5y - 2) - 2(5y - 2)\).

You'll notice \(5y - 2\) is a common factor. By factoring out this common binomial, you get \((5y - 2)(5y - 2)\), simplifying the original trinomial into a perfect square.

Algebraic Expressions

Algebraic expressions are mathematical phrases involving numbers and variables that use operational symbols. When working with algebraic expressions, especially polynomials, factoring simplifies complex expressions into manageable parts.In the expression \(25y^3 - 20y^2 + 4y\):

• The terms are manipulated through various factoring methods to express them in simpler forms.
• Factoring techniques, such as identifying the GCF or trinomial factoring, make operations like solving equations, simplifying fractions, and evaluating expressions much easier.

Understanding these techniques is crucial as they form foundational skills in algebra, helping you see deeper relationships between numbers and variables.