Problem 20 - A Graphical Approach to College Algebra

Question

Factor each trinomial completely. $$6 a^{2}-48 a-120$$

Step-by-Step Solution

Answer

The fully factored form is \(6(a - 10)(a + 2)\).

Step 1: Identify the greatest common factor (GCF)

First, identify the greatest common factor of all terms in the trinomial. The terms are: \(6a^2\), \(-48a\), and \(-120\). The GCF of the coefficients \(6\), \(48\), and \(120\) is \(6\).

Step 2: Factor out the GCF

Factor out the GCF \(6\) from each term:\[6(a^2 - 8a - 20)\]

Step 3: Rewrite the quadratic trinomial

Consider factoring the expression \(a^2 - 8a - 20\). We look for two numbers that multiply to \(-20\) and add to \(-8\). These numbers are \(-10\) and \(2\).

Step 4: Factor the quadratic trinomial

Use these numbers to rewrite and factor the trinomial:\[a^2 - 8a - 20 = (a - 10)(a + 2)\]

Step 5: Combine the factored terms

Plug the factored quadratic back into the expression with the GCF:\[6(a - 10)(a + 2)\]

Step 6: Verify the solution

To verify, expand the factored form and ensure it equals the original trinomial:\[6(a - 10)(a + 2) = 6(a^2 + 2a - 10a - 20) = 6(a^2 - 8a - 20) = 6a^2 - 48a - 120\]. This confirms our factoring is correct.

Key Concepts

Greatest Common Factor

The greatest common factor, or GCF, is an essential concept when factoring trinomials. It represents the largest number that can divide all terms in a polynomial without leaving any remainder. Understanding the GCF is crucial for simplifying expressions in math. By identifying and factoring out the GCF from a polynomial, the expression becomes simpler, making it easier to work with further. To determine the GCF:

• Look at the coefficients of each term.
• Find the greatest number that divides these coefficients evenly.

In our given trinomial, \(6a^2 - 48a - 120\), the GCF of the coefficients \(6\), \(48\), and \(120\) is \(6\). Factoring out the GCF simplifies the expression to \(6(a^2 - 8a - 20)\). Factoring out a GCF not only simplifies the trinomial but also prepares it for further methods like factoring the remaining quadratic expression.

Quadratic Expressions

Quadratic expressions are polynomials of degree two, typically in the form \(ax^2 + bx + c\). They are called "quadratic" because the highest power of the variable is squared.These expressions often need factoring to solve equations, simplify expressions, or even help in sketching graphs. When you factor a quadratic expression, you're essentially expressing it as the product of two binomials. To factor quadratic expressions like \(a^2 - 8a - 20\), you look for two numbers that:

• Multiply to give the constant term (\(-20\) here).
• Add to equal the linear coefficient (\(-8\) here).

In this expression, the numbers \(-10\) and \(2\) do the trick. They multiply to \(-20\) and sum up to \(-8\), allowing us to write the expression as \((a - 10)(a + 2)\). Understanding quadratic expressions and their factorization helps in unraveling what solutions might be hidden within a trinomial.

Factoring Techniques

Factoring techniques are strategies used to break down polynomials into simpler components. These techniques rely on recognizing patterns and applying algebraic principles.The initial step is often identifying the GCF, which we factor out first. Following this, if the remaining expression is a quadratic trinomial, we use techniques specific to these polynomials.For example, after factoring out the GCF from \(6a^2 - 48a - 120\), the next step was to factor the quadratic \(a^2 - 8a - 20\). We looked for two numbers multiplying to the constant term and adding up to the middle coefficient.Such a method is part of a larger set of techniques including:

• Grouping: involves rearranging and grouping terms to find a common factor.
• Special patterns: recognizing squares or cubes, and perfect square trinomials.
• Trial and error: systematically testing factor pairs that meet the required conditions.

By combining these techniques, factoring becomes a powerful tool, turning complex problems into simpler, more manageable pieces.