Problem 61 - Algebra A Combined Function

Question

Factor each trinomial completely. Some of these trinomials contain a greatest common factor (other than 1). Don't forget to factor out the GCF first. $$ 5 x^{3} y-25 x^{2} y^{2}-120 x y^{3} $$

Step-by-Step Solution

Answer

The trinomial factors to \( 5xy(x - 8y)(x + 3y) \).

Step 1: Identify the Greatest Common Factor (GCF)

Start by identifying the greatest common factor of all the terms in the trinomial. The coefficients are 5, 25, and 120, and the variable parts have the lowest power of each factor, which are: \(x\), since all terms have at least one \(x\), and \(y\), since all terms have at least one \(y\). The GCF of the coefficients (5, 25, 120) is 5, so the GCF of the trinomial is 5\(xy\).

Step 2: Factor Out the GCF

Factor out the GCF from the trinomial. Divide each term by the GCF (5\(xy\)):\[ 5x^3y - 25x^2y^2 - 120xy^3 = 5xy(x^2 - 5xy - 24y^2) \]

Step 3: Factor the Remaining Trinomial

Now, we need to factor the quadratic trinomial \(x^2 - 5xy - 24y^2\). We look for two numbers that multiply to \(-24\) (the product of the constant and back term \( -24y^2 \)) and add up to \(-5\) (the middle term coefficient).The numbers \(-8\) and \(3\) satisfy these conditions. Thus, factor the trinomial as:\[ x^2 - 5xy - 24y^2 = (x - 8y)(x + 3y) \]

Step 4: Write the Completely Factored Form

Combine the factored GCF with the factors of the trinomial to write the expression as a product of its factors:\[ 5x^3y - 25x^2y^2 - 120xy^3 = 5xy(x - 8y)(x + 3y) \]

Key Concepts

Greatest Common Factor

Finding the Greatest Common Factor (GCF) is a crucial step when factoring polynomials. A GCF is the largest factor that divides two or more numbers or expressions without leaving a remainder. Let's break it down with an example. Consider the trinomial expression:

• 5x\(^3\)y
• -25x\(^2\)y\(^2\)
• -120xy\(^3\)

To identify the GCF, focus on the numerical coefficients and the variable parts separately. The coefficients here are 5, 25, and 120.

• The GCF of these numbers is 5, as it is the largest number that can divide each of them.

Next, look at the variable parts:

• All terms contain at least one **x** and one **y**.
• So, the GCF for the variable part is xy.

Putting it together, we find that the GCF of the entire expression is 5xy. By factoring the GCF out of the original expression, we simplify the task of working with the polynomial, setting a solid foundation for further factorization steps.

Quadratic Expressions

Quadratic expressions are polynomial expressions where the highest degree of the variable is a square. In our given trinomial after factoring out the GCF, we are left with a quadratic expression: \[ x^2 - 5xy - 24y^2 \]Quadratics are commonly written in the form ax\(^2\) + bx + c. Here:

• a is the coefficient of x\(^2\), which is 1.
• b is the coefficient of x, here represented as -5y.
• c is the constant term with y variables, which is -24y\(^2\).

Understanding these components helps to identify how to split or rearrange terms for further factorization scenarios. In our quadratic trinomial, we look for factors of ac (product of a and c = -24y\(^2\)) that sum up to b (-5y). This process involves logical reasoning and practice but greatly simplifies polynomial factorization tasks once mastered.

Polynomial Factorization

Polynomial factorization involves breaking down a polynomial expression into the product of simpler polynomials. For the trinomial given after factoring out the GCF, we have:\[ x^2 - 5xy - 24y^2 \]Through factorization, we identified numbers -8 and 3, which:

• Multiply to give the product of ac, which is -24y\(^2\).
• Sum to give b, which is -5y.

Thus, the expression can be rewritten as \[ (x - 8y)(x + 3y) \]The factored form of this quadratic means that each part

• \((x - 8y)\) and \((x + 3y)\)

represents a root or solution to the equation that sets the expression to zero. Polynomial factorization not only simplifies expressions but is also a fundamental tool in algebra for solving equations. A clear grasp of these concepts ensures a strong foundation in algebra, making more complex mathematical problems easier to solve.