Problem 37 - College Algebra

Question

Factor each trinomial, or state that the trinomial is prime. $$ 6 x^{2}-5 x y-6 y^{2} $$

Step-by-Step Solution

Answer

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

Step 1: Identify the coefficients

The coefficients in the given expression are \(a = 6\), \(b = -5\), and \(c = -6\). Now, the product \(ac\) is \(6 \times -6 = -36\).

Step 2: Find factors

We need to look for two numbers that add up to \(b (-5)\) and when multiplied give the product \(ac = -36\). These two numbers to satisfy both conditions are \(4\) and \(-9\) because \(4 + -9 = -5\) and \(4 \times -9 = -36\)

Step 3: Rewrite and factor

Rewrite the expression by replacing \(-5xy\) by \(4xy - 9xy\) which are our calculated factors. So, we have \(6x^2 + 4xy + -9xy -6y^2\). Now, by grouping and factoring each pair, we get \(2x(3x + 2y) - 3y(3x + 2y)\).

Step 4: Final Factorization

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

Key Concepts

Trinomial Factoring

Trinomial factoring is a crucial skill in algebra. A trinomial is a type of polynomial with exactly three terms. To factor a trinomial, you express it as a product of two binomials. This helps simplify complex algebraic expressions.

The process begins with determining the coefficients, typically denoted as \(a\), \(b\), and \(c\) in the standard trinomial form \(ax^2 + bxy + cy^2\). The formula \(ac\) gives you a clue on how to break down the middle term. For instance, in the expression \(6x^2 - 5xy - 6y^2\), the product \(ac = -36\) reveals possible factor pairs that need exploration.

Next, you need to find two numbers that both multiply to \(ac\) and add to \(b\), the middle coefficient. In our example, these numbers are \(4\) and \(-9\), because \(4 - 9 = -5\), which matches \(b\). Finally, reorganize the trinomial by substituting these numbers and grouping terms. Factor out common terms in each pair, and you have your factored trinomial.

Polynomial Expressions

Polynomial expressions consist of variables raised to different powers and coefficients. The key feature of a polynomial is that each term is a product of a constant and a variable raised to a non-negative integer power. The highest power of the variable is what defines the degree of the polynomial.

• Constant: A number without variables or powers.
• Terms: Individual parts of a polynomial separated by + or - signs.
• Coefficient: The number multiplying the variable.

In our exercise, we deal with a trinomial, which is a specific type of polynomial expression. Understanding how to manipulate such expressions by factoring is vital because it simplifies expressions for further operations or solves them in equations.

Recognizing and factoring polynomials correctly allows students to solve complex mathematical problems and aids in calculus, engineering, and sciences. Factoring provides a convenient way to break down an expression for easier manipulation or evaluation.

College Algebra

College algebra serves as a fundamental building block for advanced studies in mathematics and related fields. It extends beyond basic algebra concepts, offering deeper insights into functions, equations, and problem-solving techniques.

Topics like polynomial equations, rational expressions, and factoring are core components of college algebra. Specifically, learning how to factor trinomials is crucial at this level, as it ties into solving quadratic equations, optimizing functions, and analyzing graphs.

• Simplification: Making calculations easier in equations.
• Quadratic Equations: Use factoring for finding solutions.
• Functions: Understanding behavior and transformations.

Mastery of these concepts is necessary for success in higher education fields, such as engineering and the physical sciences. College algebra provides the tools to approach complex problems methodically, developing critical thinking and analytical skills essential for academic and career success.