Problem 38 - Elementary Algebra

Question

For the following problems, factor the trinomials when possible. $$ 6 x^{2}-54 x+48 $$

Step-by-Step Solution

Answer

Answer: The factored form of the trinomial \(6x^2 - 54x + 48\) is \(6(x-1)(x-8)\).

Step 1: Identify the GCF (greatest common factor)

First, we need to find the greatest common factor of the coefficients of the three terms. The GCF of 6, -54, and 48 is 6.

Step 2: Factor out the GCF

Now we factor out the GCF from the trinomial: $$ 6(x^{2} - 9x + 8) $$

Step 3: Factor the quadratic inside the parentheses

Next, we factor the quadratic inside the parentheses: $$ (x-1)(x-8) $$ This is possible because the product of the two factors is \(x^2 - 9x + 8\) which is the original quadratic.

Step 4: Combine the factors

Finally, we combine the factored GCF and the factored quadratic terms to get the factored form of the trinomial: $$ 6(x-1)(x-8) $$ So, the factored form of the given trinomial \(6x^2 - 54x + 48\) is \(6(x-1)(x-8)\).

Key Concepts

Understanding the Greatest Common Factor

The Greatest Common Factor (GCF) is an essential concept in mathematics, especially when simplifying expressions like trinomials. It's the largest number that can evenly divide each of the coefficients in a polynomial. To identify the GCF of a set of numbers, such as 6, -54, and 48, we need to think of the highest number that all three coefficients share as a divisor.

In this specific case, the GCF is 6, because:

• 6 can divide 6 to get 1.
• 6 can divide -54 to get -9.
• 6 can divide 48 to get 8.

Factoring out the GCF simplifies the process and makes managing larger coefficients significantly easier. It's the first crucial step in factoring trinomials into a more workable form.

Exploring Quadratic Equations

Quadratic equations are a type of polynomial that appear frequently in algebra. They generally have the form of \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. Understanding and factoring quadratic equations is a foundational skill for solving complex mathematical problems.

In our example, after factoring out the GCF, we are left with the quadratic equation \( x^2 - 9x + 8 \). The goal is to express this equation as a product of two simpler binomials, often in the form of \((x + m)(x + n)\). These binomials are such that when multiplied, they give back the quadratic equation. Importantly, \( m \) and \( n \) should add up to the middle coefficient (in this case, -9) and multiply to give the last term (8). Finding such pairs requires practice, but it's a skill that becomes intuitive over time.

Converting to Factored Form

The factored form of a polynomial is an expression composed of factors whose product is equal to the original polynomial. This transformation makes equations easier to solve and their roots simpler to identify.

For instance, once the quadratic \( x^2 - 9x + 8 \) is decomposed into the binomials \((x-1)(x-8)\), you can immediately identify potential solutions. The factored form clearly indicates the roots of the equation, which are \( x=1 \) and \( x=8 \).

After factoring the quadratic inside the parenthesis, the final factored form of the original trinomial \(6x^2 - 54x + 48\) becomes \(6(x-1)(x-8)\). This process not only simplifies solving the equation but also directly helps in applications like graphing and interpreting the polynomial's behavior.