Problem 86 - Algebra and Trigonometry with Analytic Geometry

Question

Factor the polynomial. $$ 2 a y^{2}-a x y+6 x y-3 x^{2} $$

Step-by-Step Solution

Answer

The polynomial factors to \((2y - x)(ay + 3x)\).

Step 1: Group Terms That Can Be Factored Together

First, group the terms in such a way that the first two terms can be factored together, and the last two terms can be factored together: \[ 2ay^2 - axy + 6xy - 3x^2 \] becomes \[ (2ay^2 - axy) + (6xy - 3x^2) \].

Step 2: Factor Out the Greatest Common Factor (GCF) of Each Group

For the first group \((2ay^2 - axy)\), the GCF is \(ay\). Thus, you can factor it as: \[ ay(2y - x) \].For the second group \((6xy - 3x^2)\), the GCF is \(3x\). Thus, it can be factored as: \[ 3x(2y - x) \].

Step 3: Identify and Factor the Common Binomial Factor

Notice that \((2y - x)\) is a common binomial factor in the expressions that result from the factored groups:\[ ay(2y - x) + 3x(2y - x) \]. Factor out the common binomial factor \((2y - x)\):\[ (2y - x)(ay + 3x) \].

Step 4: Final Step: Verify the Factored Form

Distribute to check if the factored result is correct:\[((2y - x)(ay + 3x)) = 2ay^2 + 6xy - axy - 3x^2 \]. This simplifies back to the original polynomial, confirming the factorization is correct.

Key Concepts

Greatest Common Factor (GCF)

The Greatest Common Factor, or GCF, is a crucial concept in polynomial factorization. It refers to the largest factor that divides two or more numbers or terms without leaving a remainder. For polynomials, finding the GCF is essential as it simplifies expressions and aids in the factorization process. In the exercise, two groups of terms were identified:

• The first group: \(2ay^2 - axy\)
• The second group: \(6xy - 3x^2\)

For the first group, the GCF is \(ay\), meaning that both terms contain this factor. So, it is factored out to simplify. Similarly, the second group has a GCF of \(3x\), allowing those terms to be reduced by extracting \(3x\). Breaking down the polynomial in such a way ensures clarity and reduces complexity during further factorization steps. Remember, identifying the GCF helps make the process more straightforward and minimizes errors.

Binomial Factor

Binomial factors are expressions that consist of two terms, like \( (a + b) \). In polynomial factorization, finding a common binomial factor can be a hidden gem that simplifies expressions significantly. Once each group from the polynomial is factored using their GCF, the expression takes the form: \[ ay(2y - x) + 3x(2y - x) \]In this expression, \(2y - x\) emerges as a common binomial factor in both terms. Recognizing this common factor is key, making the factorization process efficient. By factoring out this repeated binomial, the polynomial simplifies to:\[(2y - x)(ay + 3x)\]This step helps unravel seemingly complex polynomials and transform them into simpler factors that are much more manageable.

Distributive Property

The distributive property is a fundamental algebraic property that is essential in polynomial factorization. It allows you to multiply a sum by multiplying each term inside the parentheses by a factor outside them. Here's how it works: If you have an expression such as \( (a + b)c \), the distributive property gives us: \[ ac + bc \]In factorization, distributive property is applied backward. For instance, when we find our common binomial factor \( (2y - x)(ay + 3x) \), we ensure the expression is correctly factored by distributing:\[(2y - x)(ay) + (2y - x)(3x)\]Using the distributive property verifies that our factored expression multiplies back to the initial polynomial, confirming the correctness of the factorization: \[ 2ay^2 + 6xy - axy - 3x^2 \]Understanding this property is essential to grasp how factorization goes hand in hand with multiplication, maintaining the equality between expressions.