Problem 124 - College Algebra

Question

Suppose that a polynomial contains four terms. Explain how to use factoring by grouping to factor the polynomial.

Step-by-Step Solution

Answer

The factoring by grouping method for a four-term polynomial involves dividing the polynomial into two groups, factoring out the greatest common factor (GCF) from each group, and then finding and factoring out the common binomial if one exists.

Step 1: Identify the Polynomial

The first step is to identify the four-term polynomial that will be factored. For the sake of this exercise, let's take an example of a polynomial, such as \(ax^3 + bx^2 + cx + d\).

Step 2: Divide the Polynomial into Two Groups

The next step is to divide the polynomial into two groups. These groups can be created by simply separating the four terms into two pairs. The polynomial \(ax^3 + bx^2 + cx + d\) can be grouped into \(ax^3 + bx^2\) and \(cx + d\).

Step 3: Factoring Each Group

In this step, factor out the greatest common factor (GCF) from each group. This would result in a structure like \(x^2(Ax + B) + 1(Cx + D)\), where \(Ax + B\) and \(Cx + D\) are the resulting expressions after factoring out the GCF.

Step 4: Factor by Grouping

Look for a common binomial factor in the new expressions obtained in the previous step. If a common factor exists, factor it out. This would result in a final factored expression such as \((Ax+B)(x^2+1)\)