Problem 66 - Algebra and Trigonometry with Analytic Geometry

Question

Factor the polynomial. $$ 50 x^{2}+45 x y-18 y^{2} $$

Step-by-Step Solution

Answer

The factored form is \((5x + 6y)(10x - 3y)\).

Step 1: Identify the Greatest Common Factor (GCF)

First, we need to identify the greatest common factor of all the terms in the polynomial. Look at the coefficients: 50, 45, and 18. The greatest common factor of these numbers is 1, so there's no common factor to factor out. We can proceed with factorization without taking out a GCF.

Step 2: Breakdown Middle Term for Factorization

Next, examine the middle term, 45xy, and try to express it in a way that facilitates factoring by grouping. We need two numbers whose product is the product of 50 and -18, that is -900, and whose sum is 45. These numbers are 60 and -15 because 60 * (-15) = -900 and 60 + (-15) = 45.

Step 3: Rewrite Polynomial for Grouping

Using the numbers found in Step 2, rewrite the middle term (45xy) to break the polynomial into four terms: \(50x^2 + 60xy - 15xy - 18y^2\).

Step 4: Factor by Grouping

Group the terms in pairs: (\(50x^2 + 60xy\)) and (\(-15xy - 18y^2\)). Now, factor out the greatest common factor from each pair. For the first pair, the GCF is 10x, so we get \(10x(5x + 6y)\). For the second pair, the GCF is -3y, so we get \(-3y(5x + 6y)\).

Step 5: Factor the Common Binomial

Now, notice that both terms contain the common binomial \((5x + 6y)\). Factor out \((5x + 6y)\) to get \((5x + 6y)(10x - 3y)\).

Step 6: Verify the Solution

To ensure the factorization is correct, expand \((5x + 6y)(10x - 3y)\) and check that it equals the original polynomial. The expansion confirms the factorization: \(5x(10x) + 5x(-3y) + 6y(10x) + 6y(-3y) = 50x^2 + 60xy - 15xy - 18y^2 = 50x^2 + 45xy - 18y^2\).

Key Concepts

Greatest Common Factor (GCF)

The Greatest Common Factor (GCF) is the largest number that divides all the coefficients in a polynomial evenly. When looking at a polynomial, it's essential to first determine if there is a common factor among all the terms. Finding the GCF can simplify the polynomial, making it easier to factor further.

In our original problem, we looked at the coefficients: 50, 45, and 18. The greatest common factor of these numbers is 1, indicating that there is no larger number that divides them all without a remainder. Thus, we proceed with other factoring methods as there is no GCF to factor out.

Understanding the concept of the GCF is crucial. It helps in simplifying expressions and reducing the size of numbers involved in calculations. Always start by checking for the GCF to potentially simplify the processes that follow.

Factor by grouping

Factoring by grouping is a method used to simplify polynomials by grouping terms with common factors, making it easier to factor further. This method is particularly useful when a polynomial has four terms or when it can be manipulated into having four terms.

In the given problem, we initially have three terms. We needed an approach to reframe these into four terms. By carefully analyzing the middle term, 45xy, we can rewrite it as 60xy - 15xy. This breakage allows us to then group the terms into:

• (50x^2 + 60xy)
• (-15xy - 18y^2)

Each of these grouped pairs has a common factor:

• For 50x^2 + 60xy, the GCF is 10x, resulting in 10x(5x + 6y).
• For -15xy - 18y^2, the GCF is -3y, giving us -3y(5x + 6y).

With both groupings sharing the common binomial factor of (5x + 6y), we can factor this expression out, leading us towards the final factorization.

Binomial factoring

Binomial factoring involves identifying and pulling out a common binomial factor from a polynomial. Often, after using factor by grouping, you will find a common binomial that you can factor out to simplify your expression.

In our polynomial example, both groups, (50x^2 + 60xy) and (-15xy - 18y^2), resulted in expressions containing the binomial (5x + 6y). This repetition makes it apparent that this binomial is a common factor.

The beauty of binomial factoring is illustrated when combining the two terms, already factored by grouping, as follows:

• From 10x(5x + 6y) and -3y(5x + 6y), we extract (5x + 6y) from each group.
• This results in the elegant expression: (5x + 6y)(10x - 3y).

By completing this process, we've effectively factored the original polynomial into simpler binomial forms. Always remember, confirming your work by expansion should lead back to the original polynomial, ensuring the factorization is correct.