In Exercises 65–92, factor completely, or state that the polynomial is prime. 6x²−6x−12

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Factor out the greatest common factor (GCF) from all terms in the polynomial. The GCF of the terms 6x^2, -6x, and -12 is 6. Factor out 6 to get: 6(x^2 - x - 2).

Focus on factoring the quadratic expression inside the parentheses: x^2 - x - 2. Look for two numbers that multiply to -2 (the constant term) and add to -1 (the coefficient of the middle term, x).

The two numbers that satisfy these conditions are -2 and 1. Rewrite the middle term, -x, as -2x + x to split the quadratic expression: x^2 - 2x + x - 2.

Group the terms into two pairs: (x^2 - 2x) and (x - 2). Factor out the greatest common factor from each group. From the first group, factor out x to get x(x - 2). From the second group, factor out 1 to get 1(x - 2).

Combine the factored terms. Since both groups contain the common factor (x - 2), factor it out to get: 6(x - 2)(x + 1). This is the completely factored form of the polynomial.

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Factoring Polynomials

Factoring polynomials involves expressing a polynomial as a product of its factors. This process is essential for simplifying expressions and solving equations. Common methods include factoring out the greatest common factor (GCF), using the difference of squares, and applying the quadratic formula for quadratic polynomials.

Greatest Common Factor (GCF)

The greatest common factor (GCF) is the largest factor that divides all terms of a polynomial. Identifying the GCF is often the first step in factoring, as it simplifies the polynomial and makes it easier to factor further. For example, in the polynomial 6x^2−6x−12, the GCF is 6.

Quadratic Polynomials

A quadratic polynomial is a polynomial of degree two, typically expressed in the form ax^2 + bx + c. To factor a quadratic, one can look for two numbers that multiply to ac (the product of a and c) and add to b. Understanding the structure of quadratic polynomials is crucial for effective factoring and solving related equations.

In Exercises 65–92, factor completely, or state that the polynomial is prime. 6x2−6x−12