Problem 49 - Introductory Algebra for College Students

Question

Use the method of your choice to factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication. $$6 x^{2}-5 x y-6 y^{2}$$

Step-by-Step Solution

Answer

The factored form of the trinomial \(6x^{2}-5xy-6y^{2}\) is \((3x + 2y)(2x - 3y)\).

Step 1: Identify the a, b, and c values

The trinomial is in the format \(ax^{2} + bx + c\). Here, the coefficient of \(x^{2}\) is a=6, the coefficient of x is b=-5 (considering 'y' as 'x') and c=-6 is the constant term.

Step 2: Factoring the trinomial

We need to find two numbers that multiply to -36 (product of a and c ((6*-6) = -36) and add to -5 (the coefficient b). The numbers that fit these conditions are -9 and 4. Therefore the trinomial is factored as: \(6x^{2}-9xy+4xy-6y^{2}\) =

Step 3: Group the terms

We now group the terms so as to perform factorization by grouping: = \(3x(2x-3y) +2y(2x-3y)\)

Step 4: Final Factored form

Considering \(2x-3y\) as a common factor,we have the factored form as \((3x + 2y)(2x - 3y)\).

Step 5: Checking the factorization using FOIL

We now check the factorization using the FOIL method. Let's multiply \( (3x+2y)(2x-3y) \) to see if we get our original trinomial back. Using the FOIL method: First is \( 3x*2x = 6x^{2} \), Outer is \( 3x*-3y = -9xy \), Inner is \( 2y*2x = 4xy \), Last is \( 2y*-3y = -6y^{2}\), Hence, the expression becomes \( 6x^{2}-9xy+4xy-6y^{2}=6x^{2}-5xy-6y^{2} \) which is equal to our original trinomial, hence verification is complete and the factorization is correct.

Key Concepts

Polynomial Factorization

Polynomial factorization is the process of breaking down a complex polynomial into simpler polynomials that, when multiplied together, give you the original polynomial. In our exercise, we dealt with a trinomial, which is a polynomial with three terms:

• The given trinomial is \(6x^{2}-5xy-6y^{2}\), where the coefficients are essential in determining the factorization.
• By comparing it to the standard form \(ax^{2}+bxy+cy^{2}\), we identified \(a = 6\), \(b = -5\), and \(c = -6\).

Factorization involves finding factors that satisfy both the multiplication and addition rules for numbers related to these coefficients:

• Find two numbers that multiply to \(a \times c\) (i.e., \(6 \times -6 = -36\))
• These should also sum to \(b\) (i.e., \(-5\))

Through factorization, we'd like to rewrite the trinomial in a product form, making it easier to handle or solve later.

FOIL Method

The FOIL method is a technique for multiplying two binomials. FOIL stands for:

• First: Multiply the first terms in each binomial
• Outer: Multiply the outer terms in the two binomials
• Inner: Multiply the inside terms
• Last: Multiply the last terms

In the context of our exercise, using FOIL to check the factorization of \((3x + 2y)(2x - 3y)\), it verifies if the original trinomial is achieved through multiplication:

• First: \(3x imes 2x = 6x^2\)
• Outer: \(3x imes -3y = -9xy\)
• Inner: \(2y imes 2x = 4xy\)
• Last: \(2y imes -3y = -6y^2\)

Adding all these products gives us the original trinomial \(6x^{2}-5xy-6y^{2}\), hence confirming the factorization is correct.

Quadratic Equations

Quadratic equations are mathematical expressions that can be presented in the form \(ax^{2}+bx+c\). Understanding how to manipulate these equations, including factoring them, is essential:

• They are typically characterized by the highest degree of \(x\) being \(2\).
• Trinomials, like our given \(6x^{2}-5xy-6y^{2}\), frequently appear in quadratic equation problems.

To solve quadratic equations by factorization, it's best to break the equation into simpler binomials, which can be managed easier:

• This often requires finding two numbers that satisfy specific multiplication and addition rules (as done earlier in polynomial factorization).
• By finding the zeros or solutions of the factorized form, we effectively solve the original equation.

Understanding the structure of quadratic equations and how to approach them gives a student a robust toolkit for problem-solving.