Problem 68 - College Algebra

Question

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 original expression \(6x^{2} -18x -60\) is \(6(x - 5)(x + 2)\)

Step 1: Factor out the Greatest Common Factor (GCF)

The GCF of the coefficients \(6, -18, -60\) is 6. Factoring that out, the expression becomes \(6(x^{2} - 3x -10)\)

Step 2: Factoring the Quadratic Expression

The quadratic equation inside the parenthesis can be factored into \((x - 5)(x + 2)\). This is done by finding two numbers that multiply to -10 (the coefficient of \(x^{0}\)) and add to -3 (the coefficient of \(x\)). Those two numbers are -5 and 2.

Step 3: Re-Write the Original Expression

Substituting the factored form of the quadratic expression from Step 2 back into the expression from Step 1, the factored form of the original expression is \(6(x - 5)(x + 2)\)

Key Concepts

Greatest Common Factor

The Greatest Common Factor (GCF) is a fundamental concept in polynomial factoring. It represents the largest number that divides all coefficients of a polynomial evenly. When factoring polynomials, identifying the GCF is often the first step. Here's why:

• It simplifies the polynomial, making the remaining expressions easier to factor.
• It reduces the complexity of the overall expression, allowing for further factoring strategies to be applied with ease.

For the polynomial expression \(6x^{2} - 18x - 60\), we first look at the coefficients: 6, -18, and -60. The GCF of these numbers is 6, because 6 is the largest number that divides each coefficient without leaving a remainder. Factoring out the GCF simplifies the original expression to \(6(x^{2} - 3x - 10)\). This step is crucial as it breaks down the problem into a more manageable form, setting the stage for advanced techniques.

Quadratic Expression

Quadratic expressions are polynomials of degree two and can often be factored into binomials. A standard form of a quadratic is \(ax^{2} + bx + c\). Factoring these expressions involves finding two numbers that multiply to \(c\) (the constant term) and add to \(b\) (the linear coefficient).

In our simplified expression \(x^{2} - 3x - 10\), you're dealing with a quadratic where the coefficients are \(a = 1\), \(b = -3\), and \(c = -10\). The goal is to find two numbers that:

• Multiply to \(-10\)
• Add to \(-3\)

The numbers -5 and 2 meet these criteria. Thus, the quadratic expression can be rewritten as the product of two binomials: \((x - 5)(x + 2)\). Recognizing this pattern is fundamental in tackling quadratic expressions, allowing you to transform and solve polynomials efficiently.

Factoring Techniques

Factoring techniques are a set of methods used to break down expressions into simpler parts or "factors." These approaches are useful for solving polynomial equations, simplifying expressions, and even solving real-world problems. Here are some common factoring techniques:

• GCF Factoring: As seen in our example, this involves extracting the greatest common factor from the polynomial.
• Trial and Error: Useful for simple quadratics where you guess and check pairs of numbers.
• Decomposition: Splitting the middle term to facilitate factoring of quadratics when trial and error are complex.

In the case of the expression \(6x^{2} - 18x - 60\), after factoring out the GCF and breaking down the quadratic, you interconnect these techniques to achieve a completely factored form, \(6(x - 5)(x + 2)\). Mastering these techniques involves practice and keen observation of patterns, essential for enhancing your mathematical problem-solving skills.