Problem 68 - College Algebra Essentials

Question

Factor completely, or state that the polynomial is prime. $$6 x^{2}-18 x-60$$

Step-by-Step Solution

Answer

The factorized form of the polynomial \(6x^{2} - 18x - 60\) is \(6(x-5)(x+2)\).

Step 1: Identify common factors

The given polynomial is \(6x^{2} - 18x - 60\). Look for any common factors. Here, each term has a factor of 6.

Step 2: Factor out common multiple

Factor out the identified common factor. The expression will then become \(6(x^{2} - 3x - 10)\).

Step 3: Factorize the quadratic polynomial

Now, factorize the quadratic polynomial \(x^{2} - 3x - 10\), we have to look for two numbers which add to give -3 and multiply to give -10. The numbers -5 and 2 satisfy these conditions. So, the quadratic can be rewritten as \((x-5)(x+2)\).

Step 4: Write down the final factorized form

Combine steps 2 and 3 to write down the final answer. The factorized form will then become \(6(x-5)(x+2)\).

Key Concepts

Common Factors

When we factor polynomials, one of the initial steps is to identify common factors. These are numbers or variables that uniformly divide each term of the polynomial. Think of it as finding the greatest common divisor (GCD) that applies to all the components.

For example, in the polynomial given in the exercise, \(6x^{2} - 18x - 60\), the number 6 is a common factor because it divides each term without leaving a remainder. Pulling out this common factor simplifies the polynomial, and it is a crucial step in the process of factorization. Finding common factors can significantly ease the complexity of the remaining expression, setting the stage for further factorization steps.

Factoring Quadratics

Once common factors are taken out, we often encounter factoring quadratics. A quadratic is an algebraic expression of degree two, typically in the form \(ax^{2} + bx + c\). Factoring a quadratic means expressing it as a product of two binomials.

The task is to find two numbers that add up to the coefficient of the \(x\) term (in this case, \(b = -3\)) and multiply to the constant term (here, \(c = -10\)). In the provided example, those numbers are -5 and +2 since \( -5 + 2 = -3 \) and \( -5 \times 2 = -10 \). These numbers are the solutions to the equation \(x^{2} - 3x - 10 = 0\) and they provide the factors of the quadratic as \(x-5\) and \(x+2\).

Polynomial Factorization

The term polynomial factorization refers to breaking down a polynomial into a product of its simplest parts, or 'factors'. This can involve extracting common factors, factoring quadratics or employing other methods for higher-degree polynomials.

To grasp polynomial factorization, one must familiarize themselves with different factoring techniques, such as factoring by grouping, using special product formulas, and the use of the quadratic formula when roots are not easily identifiable by inspection.

In our example, after finding and factoring out the common factor 6, the remaining quadratic was factorized, leading to the final answer: \(6(x-5)(x+2)\). It's important to check your work by expanding the factors to ensure that you get the original polynomial back.