Factoring Polynomials Using the AC Method: Simplifying (3x^2 - 15x - 12)

In this article, we will explore how to factor the quadratic polynomial (3x^2 - 15x - 12) using the AC method. Although the AC method is typically used for trinomials where the coefficient of (x^2) is greater than 1, the process still applies and will be demonstrated through detailed steps and intermediate examples.

Introduction to the AC Method

The AC method is a useful technique for factoring trinomials of the form (ax^2 bx c). The goal is to break down the middle term (bx) into two terms such that the product of the coefficients of (x^2) and the constant term equals the product of the new coefficients.

Factoring (3x^2 - 15x - 12)

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

First, we factor out the greatest common factor (GCF) from the polynomial. In this case, the GCF is 3.

[3x^2 - 15x - 12  3(x^2 - 5x - 4)]

Step 2: Identify the Coefficients

Next, we identify the coefficients of the quadratic polynomial inside the parentheses:

[a  1, quad b  -5, quad c  -4]

Step 3: Use the AC Method to Find the Factors

To apply the AC method, we need to find two numbers that multiply to (ac) and add to (b). Here, (ac 1 cdot -4 -4) and (b -5). We are looking for two numbers that multiply to (-4) and add to (-5).

[begin{vmatrix} -1  -4  1  4  end{vmatrix}]

The numbers that satisfy these conditions are (-4) and (1). Now, we can rewrite the middle term (-5x) using these numbers:

[x^2 - 5x - 4  x^2 - 4x - x - 4]

Step 4: Factor by Grouping

Now, we group the terms in pairs and factor out the GCF from each pair:

[x^2 - 4x - x - 4  (x^2 - 4x)   (-x - 4)  x(x - 4) - 1(x - 4)]

Finally, we factor out the common binomial factor ((x - 4)):

[x(x - 4) - 1(x - 4)  (x - 4)(x - 1)]

Step 5: Combine the Factored Polynomial

Now we combine the factored polynomial with the GCF we factored out earlier:

[3(x^2 - 5x - 4)  3(x - 4)(x - 1)]

Thus, the fully factored form of (3x^2 - 15x - 12) is:

[3(x - 4)(x - 1)]

Conclusion

In conclusion, the AC method can be effectively used to factor trinomials even when the coefficient of (x^2) is 1. By following the steps outlined above, we can systematically factor the polynomial (3x^2 - 15x - 12) into the form (3(x - 4)(x - 1)).

Related Keywords

Polynomial Factoring, AC Method, Quadratic Equations