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Chapter 6: Problem 49

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

Short Answer

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

Step by step solution

01

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.
02

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: \(6x^{2}-9xy+4xy-6y^{2}\) =
03

Group the terms

We now group the terms so as to perform factorization by grouping: = \(3x(2x-3y) +2y(2x-3y)\)
04

Final Factored form

Considering \(2x-3y\) as a common factor,we have the factored form as \((3x + 2y)(2x - 3y)\).
05

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

Now let's move on to factorizations that may require two or more techniques. Factor completely, or state that the polynomial is prime. Check factorizations using multiplication or a graphing utility. $$9 y^{2}-64$$

Contain polynomials in several variables. Factor each polynomial completely and check using multiplication. $$3 x^{4}-9 x^{3} y+3 x^{2} y^{2}$$

Factor completely. $$5 y^{5}-5 y^{4}-20 y^{3}+20 y^{2}$$

Contain polynomials in several variables. Factor each polynomial completely and check using multiplication. $$48 x^{4} y-3 x^{2} y$$

Graph using intercepts: \(5 x-2 y=10\)
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