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Chapter 0: Problem 74

Factor completely, or state that the polynomial is prime. $$6 x^{2}-6 x-12$$

Short Answer

Expert verified
The completely factored form of the given polynomial \(6x^{2}-6x-12\) is \(6(x - 2)(x + 1)\).

Step by step solution

01

Identify the Greatest Common Factor (GCF)

First, find out the greatest common factor (GCF) for the terms \(6x^{2}\), \(-6x\), and \(-12\). The GCF of \(6x^{2}\), \(-6x\), and \(-12\) is 6.
02

Factor Out the GCF

Factor the GCF 6 from \(6x^{2}-6x-12\). Doing so results in the expression: \(6(x^{2} - x - 2)\).
03

Factor The Quadratic Polynomial

Now the problem is to factor \(x^{2} - x - 2\), which is a quadratic polynomial. The factored form of \(x^{2} - x - 2\) is \((x - 2)(x + 1)\). The expression can finally be rewritten as \(6(x - 2)(x + 1)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Greatest Common Factor
When factoring polynomials, identifying the Greatest Common Factor (GCF) is often the first step. The GCF is the largest number or expression that divides all terms in the polynomial without leaving a remainder.
For the polynomial \(6x^2 - 6x - 12\), we examine each term: \(6x^2\), \(-6x\), and \(-12\). Notice that all three terms share a factor of 6.

To find the GCF, check the factors that each coefficient has in common:
  • 6 is the factor common to all coefficients.
  • The variable \(x\) appears in \(6x^2\) and \(-6x\), but not in \(-12\); hence it is not part of the GCF.
Therefore, the GCF for this polynomial is 6. Once identified, this factor can be 'pulled out' from the polynomial, simplifying the expression.
Quadratic Polynomial
A quadratic polynomial is a polynomial of degree 2, generally written in the form \(ax^2 + bx + c\). In this exercise, the quadratic expression is \(x^2 - x - 2\).
Quadratics are characterized by having one variable raised to the second power, meaning the highest exponent of the variable is 2. This type of polynomial typically forms a parabola when graphed.

Understanding how to manipulate and factor quadratics is vital:
  • Determine if the quadratic can be factored further.
  • Look for two numbers that multiply to give the constant term \(c\), which is \(-2\), and add up to give the coefficient of the linear term \(b\), which is \(-1\).
For \(x^2 - x - 2\), the numbers that work are \(-2\) and \(+1\), making it factorable.
Factored Form
Achieving a factored form means expressing the polynomial as a product of its factors. After factoring out the GCF and simplifying the quadratic, this exercise transforms \(6x^2 - 6x - 12\) into \(6(x - 2)(x + 1)\).
Each factor in the equation provides insight into the roots or solutions of the polynomial when set to zero.
  • The term \((x - 2)(x + 1)\) suggests the roots are \(x = 2\) and \(x = -1\).
  • Setting each factor equal to zero can help find where the original polynomial crosses the x-axis.
Factored form not only simplifies the polynomial but also makes solving for its roots straightforward.

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