Problem 41 - Algebra A Combined Function

Question

Factor out the GCF from each polynomial. $$ 5 x^{3} y-15 x^{2} y+10 x y $$

Step-by-Step Solution

Answer

The GCF is \(5xy\), so the factored expression is \(5xy(x^2 - 3x + 2)\).

Step 1: Identify the GCF

The first step in factoring out the greatest common factor (GCF) is to identify it from the polynomial. Look for the largest factor that divides each term of the polynomial. In the polynomial \(5x^3y - 15x^2y + 10xy\), each term has a numerical coefficient and variables \(x\) and \(y\). The GCF of the coefficients \(5, -15,\) and \(10\) is \(5\). For the variables, each term has at least one \(x\) and one \(y\), so the GCF also includes \(xy\). Therefore, the GCF is \(5xy\).

Step 2: Divide Each Term by the GCF

Now that we have identified the GCF as \(5xy\), the next step is to divide each term of the polynomial by \(5xy\). Performing this division gives: \(\frac{5x^3y}{5xy} = x^2\), \(\frac{-15x^2y}{5xy} = -3x\), and \(\frac{10xy}{5xy} = 2\).

Step 3: Write the Factored Expression

After dividing each term by the GCF, we rewrite the original polynomial as a product of the GCF and the simplified polynomial: \(5xy(x^2 - 3x + 2)\). Thus, the polynomial is factored completely as \(5xy(x^2 - 3x + 2)\).

Key Concepts

Greatest Common Factor

When approaching polynomials, the Greatest Common Factor (GCF) is your best friend for simplifying expressions. The GCF is the largest expression that divides every term in a polynomial without leaving a remainder. Finding the GCF involves two parts: identifying the highest common factor among the coefficients and identifying the common variables.

For example, if a polynomial includes terms with coefficients like 5, 15, and 10, the GCF of these numbers, which are all divisible by 5, is 5. Similarly, if each term has variables with exponents, such as in the expression \(5x^3y - 15x^2y + 10xy\), you'll identify the lowest power of each common variable. Here, at least one \(x\) and one \(y\) appear in every term, making \(xy\) part of the GCF.

• Find the numerical GCF of all coefficients.
• Identify the lowest exponent for common variables across terms.

Once you have the GCF, you can factor it out to simplify the polynomial and make further operations like addition, subtraction, or division more straightforward.

Polynomial Division

Polynomial division refers to dividing each term of a polynomial by a common factor, especially when factoring. Here, you apply the GCF to simplify each term in the polynomial to find the remaining expression.

In our example, once we identified \(5xy\) as the GCF, each term of \(5x^3y - 15x^2y + 10xy\) was divided by \(5xy\).
This breakdown becomes:

• \(\frac{5x^3y}{5xy} = x^2\)
• \(\frac{-15x^2y}{5xy} = -3x\)
• \(\frac{10xy}{5xy} = 2\)

Each term is reduced by canceling out the common factor, leading to a simplified polynomial. This process streamlines expressions and is a foundational technique for solving more complex algebraic problems down the road.

Factoring Techniques

Once the GCF is factored out, and polynomial division is complete, what remains is an expression that is often easier to interpret and manipulate. Factoring techniques can vary, but they all serve to rewrite the polynomial in a simpler form.
In the example provided, after dividing by the GCF \(5xy\), we were left with the expression \(x^2 - 3x + 2\).
We then wrote this as \(5xy(x^2 - 3x + 2)\). This method shows how polynomials can be expressed as products of simpler expressions.

• Use common factors to break down polynomials.
• Simplify the polynomial for easier mathematical handling.

Factoring is crucial for simplifying problems and finding polynomial roots or intercepts, making subsequent algebraic operations like solving equations or graphing functions more manageable.