Factoring Polynomials by Grouping

Every business wants to know how many products it must sell to start making a profit. Suppose a company's profit is modeled by a cubic polynomial, P of x, where x is the number of units sold in thousands.

The company reaches its break-even point when it makes zero profit and takes zero loss. So, we set the polynomial equal to zero and solve the equation.

This polynomial cannot be factored directly, so we use a method called factoring by grouping.

The first step is to split the polynomial into two groups of expressions.

We factor out a common factor from each group separately. Both groups now contain the common factor x minus 4.

Factoring out this common factor, we get x minus 4, multiplied by x squared plus 2, which equals zero.

We set each factor equal to zero. The first factor gives x equals 4, while the second factor has no real solution.

Since x represents thousands of units sold, the company must sell 4,000 units to reach its break-even point.

This example shows how factoring by grouping rewrites a polynomial into factors, making it easier to solve real-world problems.