Problem 60 - College Algebra

Question

In Exercises \(57-84\), factor completely, or state that the polynomial is prime. $$6 x^{2}-18 x-60$$

Step-by-Step Solution

Answer

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

Step 1: Identify and Factor out Commonalities

The first step in factoring a trinomial needs to check if there is a common factor that can be factored out. Each term in the expression \(6 x^{2}-18 x-60\) is divisible by 6. So, factor out the common factor 6. When you divide each term by 6, you get \(6(x^{2} - 3x - 10)\)

Step 2: Factor the Quadratic

The next step is to factor the quadratic \(x^{2} - 3x - 10\). This quadratic is such that it can be expressed as the product of two binomials:\( (x - 5)(x + 2)\).

Step 3: Write the Final Answer

Now that the quadratic has been factored, the entire expression can be rewritten in factored form. Substitute the factored form of the quadratic into the original expression to get the final factored form, which is: \(6(x - 5)(x + 2)\).

Key Concepts

Quadratic Expressions

Quadratic expressions are a special type of polynomial that specifically involve terms up to the second degree. This means that the highest power of the variable in a quadratic expression is 2. Generally, they appear in the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). Understanding the structure of quadratic expressions is crucial, as it helps in identifying how they can be manipulated or simplified. Quadratic expressions play a significant role in algebra due to their widespread use in equations, functions, and real-world applications. Manipulating these can often involve techniques like factoring, which simplifies the expression and makes it easier to solve equations or understand its behavior.

Common Factors

The concept of common factors is pivotal when working with polynomials. A common factor is a number or expression that divides each term in a polynomial without leaving a remainder. In the exercise provided, each term of the polynomial \(6x^2 - 18x - 60\) has a common factor of 6. Factoring out the greatest common factor (GCF) is usually the first step in simplifying or solving polynomial expressions. To do this, identify the largest number or algebraic expression that divides all terms in the polynomial. Once identified, you can factor it out, which simplifies the polynomial and reveals its other inherent structures. This process facilitates further manipulation, such as factoring trinomials or solving equations.

Binomial Product

A binomial product results from multiplying two binomials. Binomials are simple algebraic expressions containing two terms separated by addition or subtraction. In our stepped solution, the quadratic \(x^2 - 3x - 10\) is broken down into a product of two binomials, \((x-5)(x+2)\). Finding a binomial product involves techniques such as the FOIL method (First, Outer, Inner, Last) to expand and check multiplication results. Understanding binomial products is helpful for reversing these operations as well—factoring—where you decompose a specific quadratic product into its binomial factors, just as in the step-by-step solution.

Algebraic Manipulation

Algebraic manipulation involves using various algebraic methods to simplify or rearrange expressions and equations. In the context of this exercise, skillful algebraic manipulation is demonstrated through factoring, which rewrites expressions in a way that can be more easily understood or solved.Key techniques in algebraic manipulation include:

• Factoring out common factors, as seen in Step 1 of the solution.
• Expressing quadratic trinomials as products of binomials, used in Step 2.
• Simplifying expressions to achieve a completely factored form, resulting in the solution: \(6(x - 5)(x + 2)\).

These techniques allow you to transform complex expressions into simpler, more manageable forms, making algebraic problems easier to tackle and solve.