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When working with polynomial expressions, one of the first steps to simplify or factor them is identifying the greatest common divisor (GCD). The GCD is the highest number that can evenly divide each of the coefficients or terms of the polynomial. This step makes the rest of the factoring process easier by reducing the size of the numbers you are working with.

In the given polynomial \(80y^2 - 36y + 4\):

• Identify the coefficients and constant: 80, 36, and 4.
• Check for common divisors: Each of these numbers is even, so we begin by testing small even numbers.
• The GCD for these coefficients is 4.

After determining the GCD, you factor it out of the polynomial expression, simplifying the problem to \(4(20y^2 - 9y + 1)\). This step not only simplifies the given expression but also ensures accurate further calculations.

A quadratic expression is a polynomial of degree two. It usually takes the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are numbers, and \(x\) represents the variable. Understanding and identifying quadratic expressions is crucial because they frequently appear in mathematical problems, especially those involving factoring.

In the expression inside the parentheses, \(20y^2 - 9y + 1\), the objective is to break it down into simpler binomial factors. These factors multiply to give the original quadratic expression and are of the form \((dy - e)(fy - g)\).

You need to find two pairs of numbers whose product equals the product of \(a\) and \(c\)

• In this case, \(20 \times 1 = 20\)
• The sum of these numbers should be the middle term coefficient, -9.

By rewriting the middle term using these numbers, it can lead to easier grouping and factoring.

The method of trial factors involves considering possible pairs of factors that can satisfy the required conditions to factorize a quadratic or other higher-degree expressions. This technique is somewhat trial and error but becomes straightforward with practice and an understanding of number behavior.

In our polynomial \(4(20y^2 - 9y + 1)\), you need to:

• Identify possible pairs of numbers that multiply to a product (in the example, 20).
• These pair numbers also need to add up to the middle term coefficient (-9 in our example).

For \(20y^2 - 9y + 1\), the trial and error reveal that the numbers

• -5 and -4 satisfy both multiplying to 20 and adding to -9.

Thus, successfully factoring into \((5y -1)(4y-1)\), multiplied back out, returns the reduced quadratic, then the original expression.